katmannthree

Exactly. Every instance of the belief that elements of some particular number systems are "real" is a mapping of the human abstraction representing that number system to another human abstraction of some other system of objects that are considered to be "actually real."

The base case of that mapping is natural numbers representing possession of quantities of fungible objects. That mapping is broken by the negative integers, literally the next step up from the natural numbers. If you're concerned about the reality of the reals you can ramble on about finitism or the holographic principle and how something can possibly be real if you can't name or explicitly define every version of it, or you can recognize that every number system from the natural numbers to the complex numbers shares the property that elements of this system of numbers map to elements of physical systems and thus they are all exactly as real as each other.

Appreciate the work you do to keep this site civil. Is downweighing through moderator action a binary feature or is the pressure adjustable depending on the moderator's discretion?

The majority of advice I've received was founded in experiences 10+ years in the past, yes, and did not account for the substantial societal changes that have occurred since. My issue is not that the giver hasn't noticed changes in the world, it's that most often the underlying reason they give advice seems to be as a form of nostalgia rather than a careful consideration of what my best options are.

> By your reasoning, you would have rejected that person's advice if they had given it to you earlier because it would have looked a lot like the form of advice you have an issue with.

No, I am not saying that any advice should be immediately rejected. It should be appreciated, always, when given in good faith. It should also be silently examined, always, before application.

> It’s like imaginary numbers are fake and don’t exist

It's even better than that. Working from the other side, we generally take the natural numbers N = {0, 1, 2, 3, ...} to be "real" in a philosophical sense as we have an implied mapping of x in N fundamentally describing possession of discrete quantities of goods. This get a little more tricky when you generalize it to the integers Z = {..., -2, -1, 0, 1, 2, ...}, but we still consider these physically grounded as you can "owe" a discrete quantity of goods to someone. This is an incomplete mapping to reality as possession of y goods where y is in Z- doesn't actually describe where your y goods go to, but it's useful enough that we ignore that.

Now that's all good and well for discrete quantities of goods, but what about fractional quantities? We need a new system to describe more numbers between the numbers we already have. Thus we defined the set of rational numbers Q = {n | n = p/q where p, q are in Z and q is not 0}. This lets us compactly describe almost any number we want between the existing integers. Most still consider these to be physically grounded, because we created these to describe concrete things in our reality which they do very compactly. You can claim that at least some of these are physically grounded in reality as using 1/3 a cup of flour or buying 1/2 a watermelon is certainly something one can very clearly and explicitly do.

The rationals have some issues though, namely that they still have holes in them. Suppose you want to describe the relationship between the diameter of a circle and its circumference. The constant you use to transform one to the other, π, does not exist in Q. You can get as close as you like, but you can't actually reach it. That's a bit of a problem for people whose job it is to make sure that the things math says are correct are actually fully correct. There are other problems, suppose you want to make a rectangular plot of land whose area is 2 square miles. How long does each side need to be? You can get as close as you like by using rational numbers, but the actual length of that side (sqrt(2)) is not in Q.

To fix this, we then very delicately construct the set of real numbers R = {x | where x is in {Q and all of the numbers described above which are missing from Q}}. This is where the physical grounding of the numbers starts to get ugly, because as it turns out that just like for some numbers in Q (consider 22/7, which does not evaluate to a fixed number of digits but rather goes on forever) these are uncountable and most of them unrepresentable, i.e. if you tried to write the number out completely on a piece of paper the universe isn't big enough to hold it (and in some cases, you can't even specifically refer to the number in constructive terms as with π). This turns into a whole philosophical debate which IMO is silly but some people do take pretty seriously.

But wait, there's more! The field of real numbers is closed under addition and multiplication (and thus subtraction and division), but it's _not_ closed under some other operations. Suppose you're an EE trying to represent physical signals that very much do exist in reality, and you need to take the root of a negative real number because that is a meaningful quantity in context of the still physically grounded thing you're modeling. Well you can take the root of a negative real number, but that number is not itself a real number. Thus we must define the imaginary numbers to hold those negative roots and then the complex numbers to join the imaginary numbers back to the reals.

In every single one of these steps, the new numbers were created to describe aspects of our observed physical reality. The break from the common definition of "numbers are real because I can go buy 24 tangerines and 24 is a number" happened way back at the integers where we added a reflection around the end of the natural numbers. From a perfectly reasonable perspective then, the real and imaginary numbers are both "real" in that they describe actual physical things that exist even though you can not in fact buy -1 apples or 22/7 cats or π bananas or sqrt(-1) movie tickets.

I disagree strongly with the article's central thesis, but it is still worth critically examining the source and utility of advice. Even honest and sincerely given advice is often more a form of nostalgia on the part of the giver than a lucid examining of your specific situation and the nuances therein.

It's most often a form of "well this is what I wish I'd done when I was in a similar situation years ago" or "here's what I did when I was in your position and let me tell you how great a choice it was." That is meaningfully distinct from and a lot more common than the much more useful version "I once had something similar happen and afterwards concluded that I should have done X, but the world has changed over the decades and now it seems like kX would yield better outcomes based on Y."

What's the deal with that? Never seen a thread drop off so quickly before.

> In the past two years we've printed 80% of all US Dollars that have been printed in the history of the USD. We are not short on funding. Further it's not death or illness that is largely to blame for healthcare worker shortage as the only large drop in HCWs was on march 2020. Since then it's been growing, just hasn't recovered yet.

Yes, the issue there is sociopathic political maneuvering rather than an actual lack of national resources which could have been used to help our society in its time of need.

> Can you show me an area where the covid+ college age population was a healthcare capacity burden?

In the early days of covid it was irresponsible spring breakers spreading covid at destinations like Florida and then back home a week later. These days, look for areas with small purple cities surrounded by large red counties. As far as personal experiences go a couple Eastern states and the Pacific Northwest have issues with hospitals being literally over capacity. Many healthcare workers have been pushed past their breaking points and the national guard has been (and still is) providing manpower at the hospital next to my state's capitol (which is located next to a handful of universities). As other surrounding state's healthcare systems were overwhelmed with delta waves those states denied noncritical care and eventually sent patients out to my state. The collapse of the healthcare system is an issue for old people on vents and also a huge issue for college kids when their parents and professors up and die (caused a huge issue at one of the colleges here when a particularly irreplaceable professor passed away) and/or there are no hospital beds for them and there's a year plus waitlist on the mental health services they need now more than ever. College kids are a covid sink since they generally suffer only mildly from covid yet spread it rapidly due to their social habits and environment. The college kids are not the demographic wrecking the healthcare system, but they still spread covid and are affected by covid in their communities. Asking them to stay home to prevent spreading infections isn't that big a deal.

> Further, if we're at such risk of overburdening our healthcare system, how come we have fewer heathcare workers today than we did back in 2018?

Because we've pushed them to the breaking point rather than supporting them. Some died, some retired, some straight up left the field. Back in June of 2020 the cops had no shortage of riot gear and were always able to dig up more crowd gas, meanwhile doctors and nurses had to reuse contaminated PPE for months while a bunch of supply chain fuckery played itself out. How do you think that feels as a doctor, watching a shipment of PPE your state paid for and imported for you get confiscated by the feds and put into a big pile to get sold back to the highest bidder? How do you think it feels to notice how the police can suppress riots for months on end with endless tear gas and yet you get one N95 mask every two weeks even though you spend your entire day around people dying of covid?

> How come our hospital bed capacity is shrinking instead of growing?

See the above, we chose to stress existing resources instead of training, building, and deploying new ones. We don't have new hospitals, we don't have nationwide boot camps to get young civically minded people the skills needed to help support society.

> These don't seem to be indicators that suggest we're lacking capacity in our healthcare systems.

The indicators that we're lacking capacity are the shortage of replacement workers, the shortage of hospital beds during covid waves, the insane wait times to see specialists, and finally the countless secondary deaths caused by covid patients taking up resources that could've been used by someone close to me with once-treatable cancer whose care was delayed until it metastasized and who will now die a slow, painful, and what should have been preventable death in the next couple of years.

There are many places where that's a perfectly rational decision.

COVID was never going to be completely managed by states less than willing to take draconian measures, the purpose soft-lockdowns was instead to lighten the load on our healthcare system so that we didn't need to invest in field hospitals and a bunch of field-trained doctors and nurses. If we had done that instead not only would the optics be worse (not just many more people dying, think front page photos of them dying in muddy tents), it would also dilute the market capture of the existing entrenched healthcare system.

Yes. The simple models are simple, the fast and large and useful models are simple. We have a lot of in vitro observations which can contribute to smaller, slower and more accurate models as your linked article mentions, observations which generally are ignored as being nth order effects when people are trying to make models which have a low enough computational complexity as to be useful.

What is not missing is metaphysical quantum woo[0]. We can, have, and do observe how neurons and synapses function; the issue is that the computational complexity with current approaches is great enough to make complete neurological modeling of a microscopic worm with under a thousand total cells difficult.

[0] and even if you want to take those effects into account in your electrochemical models, all they do is turn them into stochastic models. there is exactly zero evidence of that randomness being of any real value to the thing we care about, which is the emergent macro scale properties of these systems.

Succeeded within what level of precision? The project is called OpenWorm, they have hydrodynamic models for macroscale cellular behavior and electrochemical simulations for modeling ion channels needed for the neural network. I don't have a reference handy but you should be able to find ample papers discussing accurate electrochemical modeling of synapses. The issue is that the models are complex enough that simulating a couple hundred neurons is expensive. As far as having accurate synaptic data so that they're simulating a specific worm rather than a random one pulled out of the æther, I believe that's still a work in progress.

Good talk. I agree with you (and the articles you linked which I read, BTW thank you for those there's a lot more reading for me to do) in explicit claims that the vast majority of real numbers cannot be rigorously constructed[0], and that there are many for which representation is not possible. To my knowledge this is a philosophical argument, and one for which we don't have any falsifiable theories which could be used to test one side versus another.

> You are arguing that an alternative number system should be 'infinitely dense', and I agree. But take e.g. the finite/constructive reals [1, 2], they are still 'infinitely dense'.

I'm only arguing that insofar as one can do useful things with holomorphic functions and the constructions we require to define them require the specific defining properties of the complex numbers (continuity, closure under important functions rather than clopensure, etc). If you want these properties then you're stuck with uncountable number systems and the baggage RE representation that comes along with them.

> That's exactly my point. Maybe approaching it from the point of view of 'what are the limitations of this model?' is helpful. Also see the discussion in [2]. > This is not an argument whether real numbers are useful, a good model, or interesting (there is not doubt they are all three).

Agreed, my argument is purely that the reals are "able" to "exist" in "reality" in the same way as the rationals. One of the more famous irrationals is Pi, which is of interest precisely because it is referenced by reality. The difference between the two is that the rationals are not continuous for useful definitions of continuity and so we fix that.

That "existence" is dependent on human interpretation of that word, and there's no particular reason to expect that we have the ability to see whatever the fundamental underlying reality[1] of our world even is. You could just as easily argue that negative integers do not exist because owing someone something generally requires a reference to the entity owed rather than just a indicating a lack of meaningful possession.

There are many intelligent species here on our planet which have internal models of reality, which we know are less than correct (e.g. good luck teaching anything beyond basic intuition of classical physics to a parrot), what makes us special? There are many humans who cannot handle the abstraction of charm and flavor and spin being very much real physical properties of the invisible objects underlying the reality we're able to observe.

You can argue forever over finitism and the holographic principle and what "really exists," but such discussions are fundamentally limited by the things having them. The thing we use for this analysis (logic) is itself a human construction and certainly not something which is even as real as the number 1. Our best understanding of the underlying structure of the universe right now is that it is probabilistic, which is almost antithetical to the idea of logic being the underlying set of rules by which it operates. I see the arguments people make regarding these, but what I do not see is a meaningful distinction between 1 in Z which maps to a human concept of possession of a singular instance of an object versus Pi in R which maps to a human concept of the ratio between specific properties of certain classes of objects. Both have their basis in reality grounded by human perception, both can be used to any precision you're able to, both are meaningless beyond the human constructions used to define them. The only reason we consider math to be a universal thing likely discovered by every sufficiently intelligent species is because it is of such high utility in constructing predictions which provide an evolutionary benefit.

[0] the constructability of some of the reals is likewise unimportant, they're in the set because we deemed their existence to be of future utility. there are infinitely many reals which can never and will never be specifically referenced, their purpose is no to be directly referenced but rather to be there in the background so that we can make useful assumptions concerning other numbers surrounding them. we do not have to directly reference, or even be able to directly reference for them to have value.

[1] which is under no obligation to even be the type of thing we consider to be an "underlying reality," that's just how it seems to present itself to us

I believe they’ve attempted to measure synapse weights from real worms for training the net, the firings themselves are just basic electrochemistry.

We can model neural activations but the whole-cell body simulation was still unstable last I checked (admittedly a year or so ago)

I do understand that argument, I just remain unmoved by it. Watch this, I'm about to show you a complete finite representation of an irrational transcendental number: π. That took literally three lines to represent and then an additional half page's worth that I'll skip explaining how to calculate a numerical value to however much precision you have time and space for.

Now granted, there is an underlying assumption that when you need to use that number you'll select an appropriate algorithm to compute it to the degree of precision you need, much like how if you were instead considering the rational number 22/7 you would need an algorithm to numerically evaluate it. We don't quibble about whether or not the universe has enough space to hold that one though because we have a simple abstraction which lets us refer to it with infinite precision and evaluate it with arbitrary precision. Just. Like. π. Yes, literally none of the reals would fit in the universe no matter how small you wrote them if you want to represent them with full precision. That is literally the point of the reals, that they are an infinitely dense field. It doesn't matter, we wield the same tools we used to construct them and refer to them by their names or by their construction.

If your definition of "based in reality" means "can be explicitly written out with full precision" then literally none of the reals or rationals are "based in reality" because for otherwise finite numbers you can keep padding zeros to the right of the decimal place and a finite universe doesn't have enough space to hold infinite objects. Taking a definition of reality that provides actual utility, the reals are clearly based in physical reality by virtue of their construction being explicitly guided by the objective of modeling reality. Just like the rationals and integers before them and the complex numbers after. They were literally created to model reality. Imaginary numbers are based in reality too, despite it being equally impossible to own sqrt(-2) and π melons. At best I will concede that there is an additional layer of abstraction between whatever "reality" is and what the real numbers are, but that's not a very interesting distinction given that humans are already running a dozen intermediate layers of abstraction in order to process their surrounding reality and then overlay math on top of it.

I believe we've been able to model the processes behind neural firings for quite some time now, the issue is more mapping between that micro-scale understanding of how each individual piece in a 10^10 piece puzzle with 10^15 edges affects the macro-scale emergent properties of said puzzle. It's less a lack of understanding of the processes and more a complexity so great that it exceeds the grasp of our best modeling tools thus far.

We can barely model the humble nematode c. elegans with its 330 neurons.

> on the rationals tie with reality

I feel like this ties back into the distinction between theoretical and applied math.

The basis in reality for the integers is counting discrete objects with fingers, for the rationals it's (likely) an attempt to fill in the spaces between integers using known concepts (ratios / fractions). Rationals are great if you stick to numerical work where discontinuities below epsilon can be ignored, but the rationals don't actually map to what we think of when we consider a philosophically real number system -- a discontinuous set does not match our observed experience which is that you can have any number you want between two you already have. The construction of the reals varies depending on how you want to approach it but each is equivalent: you fill in the all the holes everywhere but at the infinities so that you have a continuous closed set, just like one would intuitively expect from an infinite set of numbers representing segments of reality.

There's nothing special about the rationals which ties them more closely to reality than the reals, the rationals are just our first attempt to rigorously define all of the numbers between other numbers using the tools we had at the time.

One could just as easily construct the set $ = {x#y for all x, y in Z+} and where a#b === a + the Riemann sum of 1/(a^n) from n = 0 ... b. This also fills in some of the gaps between integers, just not enough to be interesting or particularly useful.

The rationals are interesting and stuck around because they fill in almost enough gaps to allow you to conveniently construct useful things. They're not quite there though, which is why we eventually developed the reals. And then the imaginary numbers, because despite the name physical phenomena which can be modeled using square roots of negative numbers end up presenting a compelling use case for adoption. We don't have complex numbers because some math nerd thought they were cool, we have complex numbers because they are useful in describing observed physical phenomena succinctly and as such there's enormous utility in hacking an extension onto the reals to add them.

Pulling this back around, from a theoretical perspective real & complex numbers are as real as anything else in math and are very useful to boot. You only run into issues in applied circumstances where nothing is exact and half of the things end up nondeterministic for one reason or another. Applied math requires countless shortcuts and discretionary tactics to convert things with a guarantee of correctness on the theoretical side into things which can actually be computed albeit with a correctness only within specified bounds.

Mapping between theoretical and applied math is a decent example of a pseudo one-way function, all of applied math draws from the theoretical but insights from applied math don't really map back into anything useful on the theoretical side. Which is why when we do theory and build models, we use theoretical techniques since the ability to prove correctness is the entire point. If you need numerical computation you must in exchange give up absolute correctness, which is why it is only appropriate to use during numerical computation.

> This is an observation on our interface with reality and not on its true nature, which may or may not admit reals (although my non-serious guess is that black holes form whenever you try to do something that requires a proper real number).

Well, let us know if you're able to develop a falsifiable experiment one way or another. That is definitely an interesting theory, unfortunately nobody has been able to figure out a way to poke that particular are-the-numbers-real-or-just-made-up bear.

Right, this is essentially why theoretical and applied mathematics are separate branches. Applied techniques like you described make theory incredibly cumbersome (and, importantly, not better). My argument here is not that applied mathematics is inferior (it’s what my degree is in), just that it’s generally not a good idea to carry applied techniques back into theory.

You start with doing something the most correct way possible on paper and then convert that into the fastest possible method within your allowable bounds on precision and/or convergence. Operational reordering to keep additions in floats with similar exponents is great but you save that concern until it’s time to crunch numbers. When you’re trying to build an entire theory on how something complex works you’ll have a much better time using the available abstractions to manage complexity without getting bogged down in implementation details.

Edit: addressing your point more directly, numerical computation itself must necessarily be done over fixed precision numbers but the tools we use to decide what and how to do that computation come out of theory done over the reals because of those specific properties of the reals. You can make things work over the rationals but the theory is tedious and the results of generally lower utility.

Yes, you can generally find ways to manage life with limited precision. Doing so is useful enough that it’s an entire field (applied mathematics). For many cases though it is a better choice to stick with these particular carefully designed constructs (real, imaginary, and complex numbers) because they are easier to deal with and help keep you from getting bogged down in avoidable numerical tangles. Managing error accumulation is a huge deal for some types of simulations and every little bit counts, to the point of carefully ordering your operations to minimize precision loss with the widest numbers you can afford to use.

You’d want to generalize that to a beam whose length is a significant portion of the width of the universe (at which point you should also consider relativistic effects, so there’s more math you’ll need to define over the rationals), but even so that’s not addressing the issue at hand which is that you still have to propagate your uncertainty through each calculation. Depending on the function(s) and time steps your uncertainty can quickly blow past the threshold of utility (e.g. in the case of orbits, your uncertainty could end up being numerically greater than your ability to deal with it [meaning your orbital prediction for N bodies after T time has passed is so imprecise that your craft is not capable of intercepting at all potential states]). This is the power of the real numbers, being able to bypass the accumulation of error in some cases.

That’s not isomorphic to the discussion at hand though, which refers to modeling systems. Being able to physically construct an object within epsilon does not imply that you won’t run into precision issues when modeling said object at the same degree of precision.

In other words, cutting your beams to +/- 1/2” may work for each individual beam in a building but that does not imply that your building as a whole can tolerate an average beam length being +.499” above nominal.

Can you? Most likely. Should you? You’ll need to reprove more than a handful of theorems, and for what? What advantage does using rationals instead of reals get you?

You might enjoy taking courses in real & complex analysis, the general purpose of which is to impart upon the receiver an understanding of why we’ve constructed those particular number systems and how despite the names they both describe things which are perfectly real in the philosophical sense.

Edit: recall that the rationals are simply defined as the set of numbers which can be represented in the form a/b where a is an integer and b is a nonzero integer. This isn’t some deep philosophical tie to an underlying reality, it’s just our first attempt at defining more useful numbers that lie between other useful numbers we already invented (the integers et al). The real and complex numbers are literally just the extension of that process, filling in holes between useful numbers with more numbers until the set is closed (i.e. there are no more holes, every operation between members of the set results in another member of the set). Closure, the real reason we care so much about the reals, is just a surprise tool that helps us later. For anyone who made it this far: if you find any of this interesting you should find a book or lecture on analysis. It’s not particularly difficult and presents deeper insights into the math you likely already learned.

To play devil's advocate, the apps being simple to the point of pain for power users is almost the point of the distro. If you want a distro that you can wrangle into exactly what your vision is, you'll love arch. If you want the simplest possible mostly-functional option where you don't need to change anything to get it to work, elementary isn't too bad. That said I hate using it and generally dislike their entire philosophy of taking an open program, stripping out three quarters of the code, and then calling it their own and begging for money to keep doing that. Feels gross.

None of the issues you mentioned are really fixable. Online streaming is a bit of a pipe dream for linux, legal(ish?) solutions are going to be hacky and break constantly since nobody has the negotiating power to work out deals with labels. Same with videos, alphabet is not going to just allow random apps to bypass the entire point of youtube (serving ads to eyeballs and tracking said eyeballs to serve more ads). Maps does suck, and that's mostly and openstreetmap issue. You could integrate google maps but those API keys are not cheap.

Well hello there fellow fan of that series.

Information requested: Thoughts on the big oft-theorized thing in the last book and how it’ll affect the story going forward.

I'm guessing that's because both the actual number is very low relative to the number of doses given out and that fewer people have died from the vaccine than would die as a consequence of fearmongering by a particular political party here were there open discussion of that. There are a handful of topics like this where there seems to be near-universal agreement that the best thing for society is to just not poke the bear.

It's serious, the person you replied to is talking about how to revive a dying _thing_. Whether you want to call it a spec or a brand or a product does not really matter in this context, RSS has elements of all three. Their point is that trying to get people to change their minds on RSS is going to result in wasted effort, and that if you want RSS to live you need to market the hell out of a functionally equivalent replacement to the spec that appeals in some way to the relevant decision makers.

Slow scan TV is an analog method for sending still images, not video. The fastest method allows sending a 160x120 greyscale image in about 8 seconds using ~3 kHz bandwidth.

Automakers more or less started the shortage, as a result of that everyone is in squirrel mode buying up what they can because they rightly fear having to reengineer and revalidate every single production run. That our new economy has a greater thirst for chips just compounds the issue.

Think of the statement as “backwards time travel existing would break our model of causality which has so far held up to observation.”

It’s essentially Occam’s razor. Of course we could currently be wrong, but the entire point of the western scientific mindset in which this article is written is that we can be reasonably certain that in this case we’re not. If you want to dedicate your efforts to showing otherwise, have at it. No one will stop you. They probably won’t fund you though, and on the minuscule chance you end up being right unless it’s blindingly obvious it’ll take a while to convince people (we still haven’t convinced everyone that the standard model matches experimental data as well as it does, and that’s more or less been settled for how many decades now?)

Fascinating, thank you! I imagine this would make it even more likely that large orgs would ban the use of VS Code.

How exactly does Copilot not open Microsoft up to significant legal liability, when it has been demonstrated that copilot will regurgitate entire blocks of scanned code?