Relevant, interesting problems that we have some immediate hope of making genuine work on might be, if not finite, quite difficult to produce. And it's also plausible that AI will not do as good a job of producing these as it does at solving them.
The other problem that Tao identifies is that math has typically been an unusually open subject in many respects. This openness may not work if big AI labs can afford to throw $X million at a problem to scoop you if the rumor gets around that you think you have something promising. Hence, less collaboration, and less chance of identifying these exciting new problems, infinite though they may be.
That's fair, but at least the chart has an axis. :) Since openai just released astra, I was more surprised that they would publicly show any gap to their (presumably SOTA) internal model.
BTC Liquid Network is not decentralized and does not purport to be. It's a federated sidechain, that is... it's a blockchain that runs alongside the Bitcoin blockchain (using a two-way peg that lock real BTC on the Bitcoin blockchain and issues an equivalent amount of Liquid BTC on the Liquid sidechain) but blocks can only be added to the Liquid Network sidechain by some of the handpicked members of the federation.
You are conflating the original BTC network and a lot of the other projects in the cryptocurrency / token / stable currency space.
Every time there was a new token that was 80% reminded or was governed by a central company, the original cryptocurrency enthusiasts cried fowl.
Most people don't read the fine print, don't read the founding white papers, and don't care about the differences between the protocols and the networks when they should.
I think it's more that the 'original cryptocurrency enthusiasts' tend to look the other way, because the more of these weird shitcoins/nfts/networks get minted, the more their numbers go up.
It's 2026. Show of hands, who here actually uses any of this, and why?
Got to the place where he says "As you can see, this is identical to the d/dx() operation except that the result is not divided by dx."
What does it mean with his d() operator to "divide by dx"? All of a sudden it seems like he has changed dy/dx from unfortunate notation that sort of looks like a division into something that actually is dividing two meaningful things, dy and dx? And so what the hell are dy and dx?
Actually Leibniz invented the modern dy and dx notation and did view the differentials as genuinely nonzero, which is generally speaking a safe assumption. In other words, dy/dx really was a quotient, albeit of really tiny values (at least we assume dx can become arbitrarily small while remaining nonzero). The calculation Leibniz would do looked something like this. First he would consider an equation y = x^2. Then he would consider a nonzero difference so something like y + dy = (x + dx)^2 = x^2 + 2x dx + (dx)^2. At this point he would use his starting equation to subtract y from the LHS and x^2 from the RHS, leaving: dy = 2x dx + (dx)^2. Then he would divide by dx leaving dy/dx = 2x + dx and since dx is infintisimal, he would just lop it off. Suffice it to say, just ignoring the nonzero dx on the RHS, or casting it to 0 while conveniently ignoring the division by 0 on the LHS, was rather disturbing to many critics. A lot of work had to be done by Riemann, Cauchy, and Weierstrass over the following century after Newton and Leibniz invented calculus to answer the question you are asking.
I think the best way to understand this is that division by dx is always allowed. It is genuinely a nonzero quantity. Later, we think more in a more abstract way in a tiny neighborhood around (x,y), considering what happens to dy as dx becomes arbitrarily small, but it never vanishes entirely. That explains why we can say dy = 2x dx or dy/dx = 2x and both are completely true and reasonable. I think the author's argument is that d() is a bit easier to understand because we aren't dividing by dx but it makes no sense (to me) that way. If you cannot divide by dx, a nonzero number, then why not? And if you can, why doesn't dy/dx involve zero division, which is clearly not well-defined? I think answering those questions makes calculus a lot easier to understand and that they are in a sense the hardest questions. The notation this author uses doesn't really illuminate those points and the fact that the author realizes that he is basically teaching the students to accept zero division for most of the year suggests he is basically saying we should go back to a Leibniz-era approach to calculus. I would rather make rigorous what is meant by dx/dy and what exactly dx and dy are.
Personally I dislike hyperreals for the reason that we lose the all-powerful Archimidean property (for all real numbers x, there exists a natural number N such that N > x or N = x). But yes, hyperreals are completely consistent. Robinson showed, I want to say in the 1960s, that any statement about the logical consistency of the hyperreals is true if and only if the same is true for the reals. Then hyperreals basically save Leibniz by introducing the standard part function st(), which is rigorously defined.
Again, for me, this seems a little costly. If cookbook calculus is a struggle, the student probably won't take real analysis. And once you get to analysis, the fact that the reals are Archimidean grounds the whole endeavor on a very intuitive basis. By contrast, the hyperreals are not Archimedian so we can find omega bigger than every natural number. I don't personally feel like that helps intuition and indeed requires development of non-standard analysis but I admit it is pretty cool that Leibniz-style computations can be salvaged this way.
I think the cost is just a question of tradition what you find harder and the point author makes that it's easier to learn trough non-traditional way.
I see losing Archimedean property as more of a win than loss. It's a one simple thing that turns calculus operations into single algebraic evaluation, where you can avoid dynamic limits. Things become simpler and easier to prove.
Dual numbers are basically big/little O notation. They combine beautifully with Robinson's NSA to give the most 18th century-like approach to deriving integral/derivative formulas that I know. And it's fully rigorous!
> Students have to memorize a diversity of processes for essentially performing the same task.
Is that true for differentiation ? I don't recall having to memorize many things, just how differentiation composes over +,-,×,÷, function composition and the differential of a few standard forms.
Symbolic integration, on the other hand, is a whole can of worms.
> And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?
-- George Berkeley, namesake of UC Berkeley, in 1734, critiquing infitesimal approaches to calculus.
Math uses limits because "dx" as a concept is hard to define and relies on faith that such an object can exist. It behaves as zero when convenient yet is non-zero when that breaks math. Limits have a more rigorous footing.
Your comment was correct 100 years ago. But today it is highly inaccurate.
Initially Calculus was developed using the ideas of infinitesimals throughout, although this was not yet fully formalized. The first to ground with mathematical rigor was O believe Cauchy with the epsilon-delta definition of limits. For historical reasons this caught on and is the standard way we introduce students to the subject till today.
But since then we have already discovered fully rigorous and zero faith ways to define and work with limits: Robinson's non standard analysis and using nilpotent infinitesimals a la synthetic differential geometry.
These provide completely rigorous way to view all the classical intuitions that initially develop the subject and are much easier to work with than the current standard epsilon-delta gymnastics.
Unfortunately, mathematicians are very conservative and we tend to stick with the conventional way of doing things way more than we should.
In fact a huge part of mathematical community have not fully engaged with the beautiful way of defining and using infinitesimals for calculus, even though it would greatly aid the students learning and intuition and solve the disconnect when working with physics using infinitesimals.
Infinitesimals don't rely on faith any more than any other mathematical idea. The "normal" calculus sequence being based on limits is solely due to the fact that limits were the first discovered method for rigorously formalizing the subject. Infinitesimals do a better job representing how most people intuitively think about calculus though, as evidenced by being the vehicle through which the entire field was discovered in the first place, and so now that they too rest on a formally rigorous foundation we should probably consider rebuilding calculus education around them.
They are differentials. https://en.wikipedia.org/wiki/Differential_(mathematics) has some info but is not great as a beginning introduction. dx is an infinitesimal bit of x, and dy is an infinitesimal bit of y. dx here is the same dx as in an integral, which you can think of as the width of one of the infinite infinitesimally thin rectangles whose areas are being added up to find the area under the curve: https://en.wikipedia.org/wiki/Riemann_integral
dx and dy are the smallest change ever in a function. Think about the speed of something
(that's the reason of derivatives, you can't trace the speed of a stopped car in a track, you can't divide by zero, of course, there's no motion).
No offense to you, but we had to do these kind of exercises in grade school on days when a sub was in. You'd get together in a group and work through the logic puzzle on a structured form (i.e., it let you represent what you know about the person wearing green shoes, or the person from Morocco, etc.)
I'm familiar with the format, I grew up doing these, published monthly under the name "Logic Problems" in the UK.
My issue with this particular puzzle was the ambiguity of that particular clue, it's as if it got re-written or badly translated, because clue 13 as written carries no information.
>my general understanding is that the unions are protectionist dreaming of things that were true before but not today, but they want to keep things as they were.
This mindset really baffles me, as a unionized guy with a white collar email job. Don't you, like, want things that your employer doesn't want to give you? Job protection? Better than CoL wage growth? Work from home? Internal job progression? Clear, understandable job factors agreed upon in advance?
The union negotiates for what its members want. There's nothing magical about unions that mean, like, a white collar union is going to be negotiating for all its members to get a new hard hat every year, just because that's what auto worker unions do.
I don’t “understand it” because it’s an antagonist relationship, us vs them. How’s that solved anything ever without violence?
I don’t know what the solution is, we’ve tried a few. Perhaps there are none.
Do you want to keep trying for your union to earn their seat at the table each year proving their worth, or “else”?
There is going to be another meeting where the bosses from each side will sip martinis and introduce their spouses and then they depart the galla in long black limos. It’s been written.
P.S. Your employer, your chosen object of warship, your partner or children, owe you nothing. Gotta find a way to reconcile that.
>What’s a union going to do if (doing a lot of work there) AI significantly reduces the demand for software engineers or tech workers generally?
Aside from general job protection stuff, the general answer would be that they can ensure that some of the productivity gains go to employees rather than employers. Let's say the company has been so much more efficient thanks to AI that they earn an extra 100 million in revenue with the same level of spending. So, the union goes into contract negotiations asking for a 50 million increase in salaries per annum, and employees get part of that windfall.
In a world where AI heavily disrupts software, a union striking over economics (to get an extra $50M paid out to union members every year) would immediately notice all the replacement tech workers (who were out of work due to AI disruption) streaming in to do the work.
It's really hard for me to imagine the conjunction of significant disruption from AI and effective union bargaining.
I said contract negotiations, you said strike. Even in this example, and there is an army of scabs outside the door, if a strike and its associated costs would cost the employer 75 million, it would be rational to give the union its 50.
Anyway, I agree with you that simultaneous productivity gains and deskilling as a result of AI efficiency, if it comes to pass, would work against labor power. The claim is that unions would hold no purpose under such an arrangement. I'd say if anything they'd be more important. The question is, would an employee rather have union protections if this comes to pass, or not?
Understanding the dynamics and likely limits of what those “union protections” would be critical to answering your last question.
Contract negotiations are back-stopped by strikes. If a union is asking for an extra $50M per year for their members, they’d better have a credible threat of a strike to back that.
I think most every employer would much rather fade a $75M strike expense than agree to the $50M every year expense. It breaks even in only 18 months on a cash basis, plus gives them additional union-undermining value in a very public display.
Isn't Wikimedia a non profit? Who exactly are the employers on the other side negotiating with the union here? They're higher on the hierarchical structure, but they're not actually stakeholders the same as there are in private for-profit companies.
Some of the largest unions in the world are for teachers, police, and government employees. I’m not sure why you would think a for-profit company is uniquely subject to its staff unionizing?
I have similar questions about public service unions as well. The reason why it makes sense to me for private for-profit companies is because stakeholders have a profit incentive to exploit workers. If non-profit tech workers are especially vulnerable to exploitation, then I guess it makes sense too, but I'm having trouble finding evidence for that.
>By federal law, if a company has a union, the union must represent ALL non-management workers, even those who do not pay dues.
This is not true. There can be non-union job classes even if the employer has some unionized employees. Even in different geographic areas you can have one location have class X union and another non-union (eg Starbucks).
What you cannot do, is have a unionized location with union Software Engineer I and non-union Software Engineer I. Sometimes employers try to skirt this and have non-union jobs that are too close to union jobs, and then it's lawsuit time.
2) the company alleges that the cards were signed under duress and people didn't actually want to sign. This is a necessary step in order to invoke the NLRB process
3) this puts everything on hold for the months it typically takes for the NLRB to act, allowing the company time to engage in anti-union activities.
No, it doesn't. There is no such thing as an "open shop" in the US, even under right to freeload. Union membership is determined by job classification. If you are working in a unionized environment, and your job title is a union job title, you are in the union. You will be covered by the contract your union bargains with the employer. The only question, then, is whether you pay the dues to cover the costs of union membership. In a right to freeload state, you can choose not to pay the costs.
You both have it mostly backwards. The Taft-Hartley act
mostly bans closed shops and allows states to pass “right to work” laws making payment of any agency fees to unions optional.
In no state are any employees compelled to formally join a union. In an American-style “closed shops” in a non-right-to-work state, if you don’t join the union the bargaining agreement can charge nonmembers an agency fee representing most of the dues they would pay as members. Not to mention pressure tactics to join.
Also especially in the trades in major markets unions control apprenticeship programs and hiring halls and make it virtually impossible for nonmembers to get union work, as well as gatekeeping access to apprenticeship programs in the first place.
Right to freeload laws work as I describe. Although no employees are forced to "join a union" in the sense of signing a piece of paper joining the union, in all cases they are union employees in practice. That is, the union collectively bargains their working conditions, the union supports them via grievance if the employer doesn't follow the contract, and so on. The right to freeload allows for such a person to get by without paying dues - that is, the cost of providing these collective bargaining and legal representation services.
It just struck me how dystopian the phrase "right to work" is. Like, you have the right to go down the coal mine for 16 hours a day and get lung cancer! No right to get paid, though...
“Right to work” is in the context of criminal wage theft laws, OSHA, EPA, Black Lung Benefits Act, Federal Employees Compensation Act, minimum wage (not that this affects mine workers), overtime, etc. Mine work in the USA is one of the most heavily regulated jobs there is and mine workers are very well paid. “No right to get paid” indeed. Mine workers don’t get paid when there is no mine work to do.
"For the conversion of his money into capital, therefore, the owner of money must meet in the market with the free labourer, free in the double sense, that as a free man he can dispose of his labour-power as his own commodity, and that on the other hand he has no other commodity for sale, is short of everything necessary for the realisation of his labour-power. "
The other problem that Tao identifies is that math has typically been an unusually open subject in many respects. This openness may not work if big AI labs can afford to throw $X million at a problem to scoop you if the rumor gets around that you think you have something promising. Hence, less collaboration, and less chance of identifying these exciting new problems, infinite though they may be.
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