Specifically, reverse math (a subset of metamathematics which looks at swapping axioms and theorems) allows us to show that some hard problems are equivalent to each other.
EDIT: I think this line is the most telling:
> But he cautioned that the reverse mathematics approach may be most useful for revealing new connections between theorems that researchers have already proved. "It doesn’t tell us much, as far as we can say, about the complexity of statements which we do not know how to prove."
So, at this point, it helps us understand more about problems we already understand a little about, but nothing yet about new problems.
> So, at this point, it helps us understand more about problems we already understand a little about, but nothing yet about new problems.
I don't think this caution makes any sense!
The more we learn about theorem/axiom equivalences (or more generally, the lattice of such connections) between existing proofs, the more insights we will gain into how unproved conjectures may relate to existing proofs or each other.
Only in the strictest possible sense does saying showing X tells us nothing about showing Y. Meaning a proof or identification of X is not a proof or identification of related thing Y. But that is an obviously pedantic statement.
Not to critique the person being quoted. I feel like an offhand remark may have got unduly elevated by being quoted in a "two-sides of a story" writer's dramatization reflex.
That's par for a field whose name seems to have been inspired by "reverse-engineering", which by construction doesn't try to understand products that have not yet reached the market :)
You're assuming that we know that the length of vector (a, -b) is a²+b². We don't know that.
We start by assuming that the position vector (a, -b) has length c. This implies that we can rotate that vector until it becomes the position vector (c, 0).
As you note, we can create the two vectors above from (1, 0) using linear transformation matrices [(a, b), (-b, a)] and [(c, 0), (0, c)]
So we could create the position vector (c, 0) by starting at (1, 0), applying the linear transformation [(a, b), (-b, a)], then applying a rotation to bring it back to the e1 axis.
Thus for some rotation matrix R,
R × [(a, b), (-b, a)] = [(c, 0), (0, c)]
The determinant of a rotation matrix is 1, so the determinant of the left side is 1×(a²+b²), while the determinant of the right side is c², which is how we end up with a²+b²=c².
Now the only thing which I'm not sure of is whether there's a way to show that the determinant of a rotation matrix is 1 without assuming the Pythagorean identity already.
> Now the only thing which I'm not sure of is whether there's a way to show that the determinant of a rotation matrix is 1 without assuming the Pythagorean identity already
You can define the determinant that way.
Now the question is why the cross multiplication formula for determinant accurately computes the area.
Yep - I'm just not sure if any of those proofs implicitly assume Pythagoras, and haven't thought through them properly.
I was initially going to say we know that det R = 1 by using the trigonometric identity cos²x+sin²x=1, but then found out that all the proofs of it seem to assume Pythagoras, and in fact, the identity is called the Pythagorean trigonometric identity.
Not wearing out increases the time frame across which you can amortise the costs.
If option A costs $100,000/unit and needs to be replaced every 10 years, while option B costs $300,000/unit but lasts 50 years before replacement, option B is still cheaper in the long run, even factoring in interest rates.
(You can substitute "time until replacement" in the above with "time until maintenance costs exceed the original capex", and the logic remains the same.)
Do you think GP is like, lying to you? Or maybe managers are just silly and are indeed willing to draw $500 for a pizza party but are unwilling to drop the same for a year of support for software they depend on. This is absolutely believable to me.
This is entirely jurisdiction specific, so I can't say for certain, but in almost every country I've looked into it for, there is a set of paperwork that an individual can use to independently invoice for work, without the effort of setting up an incorporated company. You will definitely need to record the income you received, and declare it on the relevant tax forms.
There is often a scale variance too - in Australia, "hobby" income is treated differently from "business" income. [0]
In Germany, there is the concept of the "Freien Berufen" ("liberal professions"), in which you can freelance without a company. [1]
> ... the client also will demand...
The client may also demand these things of you.
They are certainly capable of dealing with sole traders, and will have some services provided by people who do not have those things. (Your boss does not check if the receipt you submit for the new bookshelf for the office comes from a registered company or a sole trader carpenter.)
Depending on the scale of the services you are providing, they may prefer to deal with a registered entity, but for small one-off things, that may not be necessary.
If you are regularly working with large businesses who are funding your work, it's worth looking into the most effective tax and legal structures for you. But if you just need to send the occasional invoice off to someone who wants something quick done, it's useful to know what your options are.
One final thought - even when dealing with organisations who prefer to deal with registered businesses, you have options. You can choose to be employed by a company which does that on your behalf. Either a business which you have a good relationship with, and is willing to enter into a casual employment contract with you and bill for your services, or a dedicated contractor management company. Either way, you give up a percentage of what you bill, but in exchange, they take the paperwork and liability overhead.
> There is often a scale variance too - in Australia, "hobby" income is treated differently from "business" income. [0]
I have an ABN and I am registered for GST for side hustles beyond the hobbyist income threshold. This costs me about 10 minutes of extra admin per year when I do my tax return.
All I need to do is give the tax office three figures: How much money I earned, how much GST I charged, and how much I paid (ie how much they need to give back to me.)
"Hate the player AND the game (10 Sep 2025)" https://pluralistic.net/2025/09/10/say-their-names/#object-p...
"The one weird monopoly trick that gave us Walmart and Amazon and killed Main Street (14 Aug 2024)" https://pluralistic.net/2024/08/14/the-price-is-wright/#enfo...
"End of the line for Reaganomics (13 Aug 2021)" https://pluralistic.net/2021/08/13/post-bork-era/#manne-down
"10 Oct 2022 Antitrust is – and always has been – about fairness" https://pluralistic.net/2022/10/10/play-fair/#bedoya
And his archives for more:
https://pluralistic.net/tag/monopoly/
https://pluralistic.net/tag/antitrust/
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