There's no alternative that's significantly easier to understand and to use. The so-called "nonstandard analysis" hasn't caught on, because it's mostly the exact same arguments wrapped in slightly different language, not making them any simpler or shorter.
The language used by mathematicians is subject to constant evolution. 18th and 19th century results in analysis are not expressed and taught in the same way their original authors did it. Newton, Leibniz, Euler, Lagrange, Fourier, Riemann -- none of them expressed their results in terms of epsilons and deltas. These only caught on in the second half of 19th century, and they did so, because they were a better tool to rigorously prove the ideas.
New terminology inventions that make the subjects easier to understand take the field by storm. Some of the relatively recent examples are category theory, homological algebra, or, for that matter, the notion of sheafs, popularized by J.P. Serre himself. Mathematicians are very open to innovation, and intransigence is not the reason why we're stuck with epsilon-delta.
The reason is that nobody has yet come up with a better way of talking about these concepts. I repeatedly observe many people who seem to believe that their difficulty in understanding math stems from mathematicians gatekeeping their results. I think that this belief is just a coping mechanism. Mathematics is genuinely hard, and when people have trouble understanding something, it's easier to think that it's someone else's fault, rather than accepting one's own deficiencies.
> The so-called "nonstandard analysis" hasn't caught on, because it's mostly the exact same arguments wrapped in slightly different language
No. Let's take a nonstandard proof of the intermediate value theorem on [0,1] by Nelson.
By the transfer principle it is enough to prove this for a standard continuous function f on [0,1] with f(0)<0<f(1).
Take a finite subset of [0,1] containing every standard point. Colour its points blue, green, or red according to whether f is negative, zero, or positive at tha point.
The first point of the interval is blue and the last red. Hence either awe can find some green point, or we can find two neighbouring points that have different colours, the first blue and the second red.
In the first case there is a zero, so we are done. In the second, let the neighbouring points be p and q. By the completeness of the real numbers, every nonstandard real in [0,1] is infinitesimally close to exactly one standard real. So p and q are infinitesimally close to some standard real number, let's call it z.
Standard continuous functions send infinitesimally close points to infinitesimally close points. So f(p) and f(q) are both infinitesimally close to f(z). But f(p) is negative and f(q) is positive. The only standard number infinitesimally close to both positive and negative numbers is zero. Thus f(z) is zero. This proves the theorem.
You tell me, which standard proof is this? It's certainly not the nested interval proof. Not the supremum proof. Not the bisection proof in disguise. Which argument does it wrap in slightly different language? Can you point to a single textbook, course note or lecture that gives such an argument?
No. One could of course argue that this is not simpler/shorter than the usual arguments. But it is very different from them. Saying that it's the same arguments repackaged in a different language is just wrong, and detracts from an otherwise valid point.
This is the standard nested interval proof, you’re just replacing the limiting step of taking smaller and smaller intervals with the nonstandard way of expressing the same thing.
I'll be honest: your one sentence response tells me you did not read the proof above in any detail.
I chose Nelson's proof precisely because its construction is well-studied and well-understood. The same construction of a mesh containing all standard points, with the coloring forcing a tiny multicolored cell, extends from the interval to the triangle. In one dimension you get two adjacent differently colored points; in two dimensions you get an infinitesimal triangle whose three vertices have the three relevant colors. Taking their common standard part and applying continuity gives a short proof of Brouwer's fxied-point theorem on the triangle.
But it is well-understood (there's a whole field studying such questions [2]) that the nested interval proof of the Intermediate Value Theorem does not generalize to proving Brouwer's fixed point theorem on the triangle [1]. This fact can be derived from a computability argument as well [3].
Nelson's argument does generalize to prove Brouwer, so it's not the nested intervals argument. But really, nobody cares about these technical reasons. It's obvious to most math undergraduates that Nelson's proof is not the nested interval proof, the clear absence of any nested construction kinda gives it away. The only reason it was necessary to get technical is that you did not really inspect the proof before claiming it was nested intervals. The technical results cited above are just a formal way to show that any correspondence you might imagine between the two proofs is just not there.
[1] Shioji/Tanaka: "Fixed Point Theory in Weak Second-Order Arithmetic", Annals of Pure and Applied Logic v47, pp 167188 (1990).
What you just described is a classic proof of Brouwer's fixed point theorem using Sperner's lemma. The proof you cited earlier does not generalize to it on its own, the Sperner's lemma is a crucial combinatorial ingredient. It's crucial, because it only works on spaces with the topology of the triangle; you cannot perform the same argument on, say, an annulus.
In the standard formulation, you apply the Sperner's lemma to find smaller and smaller triangles, and apply compactness, precisely as in the standard proof of intermediate value theorem.
The rest of your post, where you quote reverse mathematics stuff, is completely irrelevant to the point I was making. Nothing I said is about what theorems follows from what axioms, but rather whether nonstandard analysis is meaningfully different, clearer, or more useful language than standard one. It is not.
You made a sweeping claim that nonstandard analysis arguments are the exact same arguments, wrapped in nonstandard langauge. I explained that (while your other claims about simplicity may be valid) this is not so and detracts from the rest of your points. I challenged you to defend your "same arguments" claim by finding any standard analysis textbook which teaches a standard language version of Nelson's argument as a proof of the IVT. Let me recap what happened since then:
1. Two comments ago you confidently claimed that Nelson's IVT proof is "the standard nested interval proof" with the limiting step written in nonstandard language. That is a straightforward claim about the structure of the proof, one that you didn't bother to substantiate, and that is straightforwardly false.
2. After I explained why it's false (Nelson's construction proves BFPT, which no nested interval type proof can do), you changed your response: now the Sperner lemma was a "crucial additional ingredient". But Nelson's combinatorial step, that opposite endpoint colors force a blue-red interval, _is_ the one-dimensional instance of the Sperner lemma (and indeed the base case when you prove Sperner's lemma for arbitrary dimensional simplices by induction; the analytic part is independent of dimension, once you find an infinitesimal multicolored simplex, you take its common standard part and apply continuity exactly as before).
3. Then you wrote this:
> In the standard formulation, you apply the Sperner's lemma to find smaller and smaller triangles, and apply compactness, precisely as in the standard proof of intermediate value theorem.
There is a standard proof of Brouwer via the Sperner lemma, and it is _also_ not of the same form as the standard nested interval proof of the IVT. In the nested interval proof, you find a sign-change interval, then find a smaller sign-change interval inside it, and so on. The intersection of all of these contains a point, and that's your zero. Nelson's proof does not do this, and neither does the standard proof of Brouwer via the Sperner lemma: you do not, and cannot, take a 3-color interior triangle, then find a smaller 3-color interior triangle inside it and so on. Even the first step would not work, since the inherited labelling does not satisfy the right boundary condition relative to the small triangle!
And this is also why the computability paper I cited ("where I quote reverse mathematics stuff" ;) was very much relevant. There can be no effective "nested triangle" proofs of the Brouwer fixed point theorem at all, because such a proof would let you compute a Brouwer fixed point, and there are examples of computable maps on the triangle without computable fixed points. If Nelson's IVT proof was the nested interval proof, then swapping in the higher-dimensional Sperner step would give a nested-type proof of BFPT. No such proof can exist. Since Nelson's argument proves the BFPT without any change to the analytic part, it is not a nested interval type argument.
You first misidentified Nelson's proof as nested intervals, and then treated Sperner as an additional ingredient even though the coloring step in Nelson's proof is already the corresponding Sperner argument. Those are both fairly serious misunderstandings about these proof. Given this, I don't think our exchange leaves readers with much confidence in your assessment of NSA's drawbacks and benefits. That's a disappointing outcome, as far as I'm concerned. There are good arguments to make that NSA adds little value to undergraduate education, such as simplicity or the difficulty of the prerequisites, and good conversations to be had about them. But "NSA proofs are the same proofs wrapped in a different language" is not one, and I wish you had just narrowed it instead of doubling down.
I never said the triangles in the standard proof are going to be nested, so your whole segue into reverse mathematics is, just like I said, irrelevant. The point of the argument is that you can find a sequence of triangles with differently colored vertices, the vertices of which all converge to the same point (thanks to compactness), which contradicts continuity of the retraction on the boundary. The nonstandard version of this is exactly the same argument, it just replaces the explicit limiting step that contradicts continuity with the an argument that uses the nonstandard formulation of continuity in terms of infinitesimals.
If that makes it easier for you to understand it, in the standard proof, you also color every point of the rectangle, with the color of the edge it retracts to (picking the colors of the vertices of the big triangle arbitrarily, just making sure that the color of each vertex is a color of one of the edges it belongs to, not one of the opposite edges). Then, an easy argument from continuity shows that no interior point will have points of three different colors arbitrarily close to it. Finally, applying Sperner's lemma as above proves that such point must nevertheless exist, obtaining contradiction with the existence of the retraction.
and then treated Sperner as an additional ingredient even though the coloring step in Nelson's proof is already the corresponding Sperner argument.
I don't understand what are you saying here. What I'm saying is that for the coloring proof of BFPT to work, whether clothed in standard or nonstandard language, you must perform a combinatorial argument that uses a topology of a triangle as a necessary ingredient, similar in shape to the proof of Sperner's lemma.
You opened with the claim that nonstandard analysis hasn't caught on because it's "mostly the exact same arguments wrapped in slightly different language". I pointed out that the arguments are in fact very distinct: e.g. Nelson's proof of the intermediate value theorem is something that any NSA student would see, but no standard textbook teaches IVT by a standard language counterpart of it.
One post later, you answered that Nelson's IVT argument is in fact the "standard nested interval proof" with the limiting step rewritten in nonstandard language.
That claim is simply wrong. Why? Because Nelson's construction straightforwardly generalises to Brouwer, while the nested intervals proofs cannot. The discussion of reverse mathematics / computability is not a tangent, it explains precisely why Nelson's proof can generalise to give the BFPT in two dimensions, whereas the nested intervals proofs (which you claim is the same) cannot.
You then brought up that the BFPT generalisation of Nelson's argument needs the Sperner lemma as "crucial additional ingredient". Now you make the same point again:
> What I'm saying is that for the coloring proof of BFPT to work, whether clothed in standard or nonstandard language, you must perform a combinatorial argument that uses a topology of a triangle as a necessary ingredient, similar in shape to the proof of Sperner's lemma.
Presumably you keep pointing this out because you think it justifies some claim like '1D Nelson is actually nested intervals with the limiting step recast in nonstandard language, even if the 2D generalization of Nelson is not'.
But it does not. The combinatorial content is the same, the 1-dimensional interval case uses the topology of the domain just as much as the 2-dimensional triangle case does. The 2D argument wouldn't work on the annulus, and the 1D version would not work on the union of two disjoint intervals. The Sperner lemma is present in 1D, and present in 2D. If instead your point is only that proving the BFPT requires a harder case of the Sperner lemma than IVT, then of course it does. But what relevance does that have to the original claim that Nelson's IVT proof is the nested-interval proof? The proof of the Sperner lemma is pure combinatorics, it does not involve any (standard or nonstandard) analysis.
Or have you changed your mind on your earlier claim that Nelson's proof is "is the standard nested interval proof"?
If so, I think that's great, and closes the thread on whether NSA is largely the same arguments, since even the first proofs of the basic results are different.
If you still think that it's the nested interval proof, well, I am not sure what else to say, apart from linking the literature which studies this exact question, that I've already done, and that you dismissed as a tangent.
Either way, this discussion went on for too long at this point, so I won't monitor it further.
The entire business model of Anduril is based on eschewing the “cost-plus” model in favor of fixed-price contracts, where your profit margin is not capped, and where you are actually incentivized to keep the costs down, instead of pumping them up.
The Rotchschilds, the Rockefellers, and the Du Ponts would have done very well in life just as well had they went to public schools, while the alumni of Baltimore public school would not become Rotschilds and Rockefellers and Zuckerbergs had they gone to Pillips Exeter.
In many things with regards to education you're not paying/applying for a better quality system in and of itself, but for better quality peers. A public school with a e.g. Exeter quality class would have phenomenal results. By contrast an average public high school class at Exeter is going to have mediocre results. We even tested this. Around COVID many top universities rolled back their testing requirements for admittance under an equity based argument. Those universities are now mostly rolling back those decisions because they found that being less selective in enrollment resulted in significantly worse outcomes. Not exactly a surprise, well at least it shouldn't be.
I don't know how the US works, but in the UK public (private...) schools have incredible facilities for sports, the arts, personal lessons and tutoring, and so on.
Meanwhile the state sector has a lot of crumbling buildings and hand-me-down textbooks.
Public school kids still manage to screw up but they're more likely to land on their feet. Access to the upper levels in careers like law and finance is very much about "polish" (cultural identity) although working class kids sometimes gain entry, often as charity cases.
At the same time public schools used to be notorious for brutal bullying. So survivors are often emotionally traumatised and have had empathy beaten out of them.
In the US public (US terminology) schools are quite well funded, contrary to what many think. We recently hit a record of just under $18k per student per year for high schools. [1] I went to school in some quite low income areas, and the facilities were great. The big issue is that much of the student body was not.
So the facilities, textbooks, and so on were regularly vandalized, stolen, or simply destroyed. But nonetheless we had a nice gym, auditorium, track/field, and so on. We even had things like laser disc players back when that was pretty bleeding edge. In modern times I'd expect to see various luxuries like 3d printers and stuff like that even at low income schools in the US, at least if they haven't been stolen or [intentionally] broken.
If I had to choose for my children between Exeter peers at a US public school, or US public school peers at Exeter, it wouldn't be even remotely close.
You are dramatically mistaken about the funding and education quality for low-income school districts. You might find some with good sports facilities, probably from a grant from an outside organization or local college, but anything related to academics, facilities, counseling… forget it. Most public school teachers have to buy their own classroom supplies That is fucked up. I’ve seen districts that don’t have enough available cash to buy food to give everybody a lunch. In one school, a teacher got very sick — they didn’t have enough money to pay a sub, and there wasn’t any other class being taught that we could have been folded into… oh well! Just sit in the library for that period, and everybody gets an A for the year.
My childhood was split between a low income district and an upper-middle-class district. Neither sets of children were likely to vandalize a goddamned thing. In the low-income district, the vibe was militaristic. We were expected to sit with backs straight and our hands folded when not writing. We moved in orderly single-file lines between classrooms and had to line up in formation outside before we were allowed recess. The education was extremely basic. The 5th grade students were barely on ‘chapter books.’ Science class was rote memorization of extremely basic physical science with no experimentation except for one class, the highlight of the year, where we each got a AA battery, two wires, ands a light bulb and had to go through a prescribed set of “experiments” where we would put the wires in various, obviously wrong configurations and finally make a circuit. Art class was still macaroni collages and music class was zero-education sing-alongs from the same song book they used in the 60s. In the upper-middle-class district we were reading Greek mythology and pulling out philosophical themes, exploring people’s emotional motivations, and exploring the historical context. Took pictures with cameras obscura that we made ourselves, made our own Leyden jars and did some very well-designed and open-ended experiments. Art was mostly graphite work from “drawing on the right side of the brain“ where we discussed things like negative space, and we made simple pottery in the school’s ceramic studio.
To imply it was the fault of the student body of that low-income school district that they might have been less engaged or have worse outcomes in life is fucking bananas. Gather more information before making these sweeping generalizations.
You're speaking of elementary which I can't say much one way or the other about in part because I remember things as a child would - I didn't have a 'mature' understanding of what was happening around me back then - I've also never looked up the data on it. I'm referring to middle/high school. And there at least, anything remotely like what you're describing would be highly atypical.
As for the data I've already cited the $18k spent per student which also leads to another common misconception. Low income schools receive more average public funding than high income schools, thanks to various redistributionary measures. Similarly for things like vandalism, 67% of schools report file at least one criminal report on property crime per year. [1] The most common rate is 3-5 per year, followed by 6-10 per year. And that dramatically understates the rates, because most things, as in the vastly overwhelming majority, are not criminally reported.
You can’t make those funding generalizations because the end amount that schools spend on students is defined locally. I just looked in the Massachusetts school funding database. The town I’m not going to out myself as having attended spent ten thousand dollars per year less per student than the Upper-Middle-Class Weston, and the fantastically expensive Cambridge spent 7k per student more than that annually.
Local taxes are where much funding comes from but the redistributionary measures come from both the state and Federal in 'normal' funding as well as a wide array of various program based funding, like Title 1 at the Federal Level. This is why looking up the total amount spent per student is a bit trickier than it might seem. This [1] paper breaks down all funding, with details by state. Check out page 23. So for instance in Massachusetts high poverty districts receive the highest public funding by an very wide margin, with $27,468/student contrasted against a statewide average of $21,767.
What your experience may reflect is more of a rural/urban divide rather than poor/rich. If you happened to live in a low income rural area with few minorities, then your public funding is going to be much less than an urban school with more minorities. You also need to account for basic cost differences, which is going to be factored into redistributionary measures. A dollar goes way further in rural areas than it does urban, which yet further reduces the amount of funding that rural areas receive. Finally schools with higher minority populations tend to have worse outcomes (higher crime, lower scores, etc) which means by the inverse you also receive less funding if you have fewer minorities.
There's even a hint of this possibility in the data. In Massachusetts, the highest poverty districts do indeed receive, by an overwhelming margin, the most funding. But the second group with high levels of 'mid poverty' receives the lowest funding amongst all 4 groups. The data doesn't specify, but I'd expect that this group is significantly comprised of rural and low minority populations.
You’re trying to make conclusions about local situations using federal funding data. No matter how many words you type about it, your approach is fundamentally inadequate and you’re making sweeping generalizations based on it. I’m out. Have a nice day.
That's not federal funding data. The paper provides specific numbers on the total funding including local, state, and federal. Looking only at local funding, as you likely did, is completely misleading.
Education is essential in general. But as far as the classic definition of "success" is concerned (usually revolving around wealth of some kind), education is like luck. It could make the difference between 2 people in otherwise identical situations and status. But education alone sadly has a vanishingly small chance of competing with "old money" and generations of accumulated wealth and influence.
There is absolutely no evidence this is true. There are always a few first generation talents who build empires from dirt and nothing (for values of nothing that usually include war, grift, and slavery).
But once an empire is established it becomes largely self-sustaining, and the inheritors get a huge head start over everyone else in their cohort.
Inherited wealth is not self-sustaining at all: if you spend it down, it disappears. Most inheritors of large windfalls will not manage to pass down that wealth in turn; it takes a rare ingrained cultural attitude to make it genuinely "self-sustaining", and even then that really means leaving the wealth as it is and managing it well - not using it to pursue any immediate "head start" in living standards over your peers.
Keeping down cost per child? What are you talking about? The cost per child has skyrocketed in the US over last 20 years, particularly among the worst-performing schools. Baltimore spends more per child than Mountain View. NYC spends more per child than Palo Alto.
There is a certain view, popular among many people, that poor performing schools perform badly because they are underfunded. It has been vigorously spread by politicians, teachers, and activists. Well, we ran the experiment. With enormous federal and state spending, we managed to mostly equalize the spending, and, in many cases, fund the badly performing schools more than the good ones. The results of the experiment are very clear: additional funding, by and large, does not *cause* significantly improved educational outcomes. Any observed correlation between school funding and outcomes is a result of who lives in the area, and who attends the school. The casual arrow does not go from funding to outcomes, but from the school population to both funding, and, in parallel, to outcomes.
Increased funding did improve educational outcomes. It just didn't improve it enough to be considered worthwhile.
The fact is that educational outcomes are caused by a large number of variables. Areas with (historically) poorer schools have less competent (or more corrupt) administrations so money doesn't go as far. Even with good leadership at the top (superintendent or public school committee), if the source of that money is not something you can rely on consistently enough to pay salaries with then you are not going to see significant changes.
Pre-K education is also critical to determining later school performance, as knowing the alphabet and numbers are practically assumed by kindergarten curriculums as they are covered far too far to be more than a review. But we have not tried that yet, so we'll see if that pans out...
Enormous federal and state spending on what, by whom, directed by whom, for whose financial benefit?
The results of the experiment aren't clear at all. Increased spending as was done here hasn't had good results. That's not an argument against spending, it's an argument for understanding why it didn't work, and what could be done about it.
We have data on this too. If, instead of spending at large, you look at spending on any specific thing (school lunches, after school classes, teacher wages, materials, classroom aids, electronic equipment, facilities, etc), you find the exact same thing: nothing we have tried so far has made any significant difference. The proponents of increased school spendings are completely blind to empirics, and their only answer to that is “we just need to spend even more”.
The point is that while power and water is scarce in space, the NY state has no jurisdiction there. It might literally be easier to build data center in space than it is to overcome social and political pressure in NY.
It's extremely unlikely that companies will find no locations that let them build zero power zero water data centers. And bulk AI stuff isn't latency sensitive either so the acceptable build radius is huge.
Most locations that could provide guaranteed power for western companies with stable governments are also full of NIMBY or NGO who could easily blockade those datacenters. Then you'll have to run to offgrid locations where you have to build 24hr battery for solar with closed loop cooling. Is 24hr battery+solar on earth cheaper than LEO satellites in constant sun without batteries? That's the question that would make or break satellite datacenters, as political climate against AI datacenters is increasingly getting bad with increasing electricity prices.
Exactly. This whole thing is not about being able to build whatever they can afford, it’s about being in a (stupid) race to build it as fast as possible.
Yes, the furniture became less durable as it became much cheaper. You can still buy durable furniture, but people just aren't interested in it for the most part.
But the durable furniture you can still buy is now several times more expensive than 30 years ago, much more than the inflation. Apples to apples, they became less affordable, not more. If you add up the cost of replacing everything every so often, the downtime caused by the replacing, and the periods where your possessions are a little broken but not enough to replace them - there's a strong argument that lower prices caused our quality of life to go down, not up.
I don’t think it’s true. I have been recently to a store here in Virginia that specializes in furniture made by Amish in USA, and while significantly more expensive than IKEA/Wayfair sort of stuff, I doubt that it was significantly more expensive than similar stuff 30 or 50 years ago.
I'm sure some smart people could come up with a SQL-esque language that is even more readable to non-technical folks than programming syntax.
And somehow, none of the thousands of very smart mathematicians have done that, or if they had, it has not seen wide adoption. I recommend contemplating on this: if math could be made easier by changing notation, why hasn't this already happened?
Momentum, mainly. Richard Feynman invented a more intuitive triangle-based syntax for sin/cos/tan/etc but eventually abandoned it and conformed with mathematical norms for the sake of ubiquity.
Gas prices have barely changed in inflation adjusted terms over last 50 years, and if anything, the trend is slightly downwards. At the same time, inflation adjusted incomes went up, and cars became significantly more fuel efficient.
Dividing profits by employee count and calling this "fair share" is ludicrous. It exhibits a childish understanding of the world and the economy. If you believe that you're cheated, and you are creating significantly more value than you're paid, start your own single-person company and sell your services to others.
The truth is that by working for a company, you get access to environment that makes you much more productive than you'd otherwise be on your own. You also are not on the hook for most of the risks. It is patently unfair, and extremely short-sighted, to claim that investors deserve no compensation for their investment.
Implying people are cheated is a nice way to increase dissatisfaction, if the goal is to produce resentment.
I'm all for unions and collective bargaining but I'm not a fan of manipulated dissatisfaction.
Access to many of the companies I worked for allowed me to support their monopolies just like desktop bundling (MS), pay store fees(AAPL), search advertising (GOOG), and so on. I guess you could call that "more productive"?
It's probably easier to understand this concept if you forget about tech, and think about something like steel mill, or a mine. If you want to make money as a worker making steel, if you just start doing it in your backyard by yourself, you'll produce very little steel in an hour of your labor. If, instead, you obtain access to enormous, capital-intensive machinery, by joining a pre-established company that owns a blast furnace, and has pre-existing business relationships with ore and fuel suppliers, the hour of your labor will be made much more productive. You will be producing a lot of value by working in a steel mill, but only because the pre-existing capital investment, and process organization will enable you to do so. Therefore, the business will contribute to your productivity in an enormous way, making the idea that the profits are entirely your own contribution just silly.
The same is, of course, true about Google or Apple. Working at Apple will make it much easier for you to be productive than working at your own company. The nice thing about our industry is that the latter, while more difficult, is actually possible -- unlike steel mill workers, software engineers don't need as much capital investment, and can run highly successful companies that employ just one or a handful of people. It's just hard and risky to try that, hence people prefer to pretend that the productivity enabled by working as part of established, successful company is entirely due to their own merit.
On the other side: without the workers, those large capital investments are useless lumps of metal.
Both Capital and Labour invest in the company, in different ways. It's not at all clear to me that the optimal arrangement is that Capital reaps the bulk of the return from those investments.
The value of a business isn't its wages or its immediate profits: it (over-simplified) represents a claim on the future profits.
Equity grants (and especially options) aren't particularly controversial in tech, and I'm not saying they're the answer to the problem at hand. But they're a mechanism for allowing workers to share in the gain from their investment. All the problems (especially with US tax treatment of options) notwithstanding.
I agree, if you think you're being cheated you should thinkg about starting your own company. Startups are much better on this. I definitely disagree with the idea that workers don't take on the risks. Workers get fired, relocated, their compensation gets changed, etc etc.
> I definitely disagree with the idea that workers don't take on the risks.
When $employer beats or misses targets, that affects my bonus (which is formalized for only the higher pay bands, probably on the theory that we can actually have some individual impact somehow) but not my base salary which is actually the vast majority of what I'm paid.
Risk isn't zero, but it's far less than what the equity holders see.
Mega corps have access to guaranteed bailout by governments as they're considered National Security concern, plus massive lobbying departments to buy all the laws they need to favor themselves over most potential competitors both external and internal, plus massive legal departments with experts at twisting the laws they didn't manage to buy yet into meaning whatever they want them to mean.
And after all that socialism-for-the-rich, any remaining risk is transferred directly to their employees in the form of lay offs, coupled with non-compete clauses that forbid the most competent, for months to years, from working at their core area of expertize, plus arbitration clauses that forbid them from seeking relief at the actual courts. Clauses that are enforced by the courts due to the aforementioned lobbying and purchased laws.
The language used by mathematicians is subject to constant evolution. 18th and 19th century results in analysis are not expressed and taught in the same way their original authors did it. Newton, Leibniz, Euler, Lagrange, Fourier, Riemann -- none of them expressed their results in terms of epsilons and deltas. These only caught on in the second half of 19th century, and they did so, because they were a better tool to rigorously prove the ideas.
New terminology inventions that make the subjects easier to understand take the field by storm. Some of the relatively recent examples are category theory, homological algebra, or, for that matter, the notion of sheafs, popularized by J.P. Serre himself. Mathematicians are very open to innovation, and intransigence is not the reason why we're stuck with epsilon-delta.
The reason is that nobody has yet come up with a better way of talking about these concepts. I repeatedly observe many people who seem to believe that their difficulty in understanding math stems from mathematicians gatekeeping their results. I think that this belief is just a coping mechanism. Mathematics is genuinely hard, and when people have trouble understanding something, it's easier to think that it's someone else's fault, rather than accepting one's own deficiencies.
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