That physical representation argument never made any sense to me. Like say I have a rock. I split it in two. Do I now have 2 rocks? So 2=1? Or maybe 1/2 =1 and 1+1=1.
What about if I have a rock and I pick up another rock that is slightly bigger. Do I now have 2 rocks or a bit more than 2 rocks? Which one of my rocks is 1? Maybe the second rock, so when I picked up the first rock I was actually wrong - I didn’t have one rock I had a little bit less than one rock. So now I have a little bit less than 2 rocks actually. How can I ever hope to do arithmetic in this physical representation?
The more I think through this physical representation thing the less sense it makes to me.
OK so say somehow I have 2 rocks in spite of all that. The room I am in also has 2 doors. What does the 2-ness of the rocks have in common with the 2-ness of the doors? You could say I can put a rock by each door (a one-to-one correspondence) and maybe that works with rocks and doors but if you take two pieces of chocolate cake and give one to each of two children you had better be sure that your pieces of chocolate cake are goddam indistinguishable or you will find that a one-to-one correspondence is not possible.
To me, numbers only make sense as a totally abstract concept.
For people who read this parent comment and are tempted to say “well of course complex numbers can be ordered, I could just define an ordering like if I have two complex numbers z_1 and z_2 I just sort them by their modulus[1].”
The problem is that it’s not a strict total order so doesn’t order them “enough”. For a field F to be ordered it has to obey the “trichotomy” property, which is that if you have a and b in F, then exactly one of three things must be true: 1)a>b 2)b>a or 3)a = b.
If you define the ordering by modulus, then if you take, say z_1 = 1 and z_2 = i then |z_1| = |z_2| but none of the three statements in the trichotomy property are true.
[1] For a complex number z=a + b i, the modulus |z|= sqrt(a^2 + b^2). So it’s basically the distance from the origin in the complex plane.
> I have a real philosophical problem with complex numbers
> I believe real numbers to be completely natural
I have to say I find this perspective interesting but completely alien.
We need to have a way to find x such that x^2-2 = 0, and Q won’t cut it so we have R. (Or if you want, we need a complete ordered field so we have R)
We need to have a way to find x such that x^2+2 = 0, and R won’t cut it so we have C. (Or if you want, we need algebraic closure of R by the fundamental theorem of algebra so we need C)
I don’t really think any numbers (even “natural” numbers) are any more natural than any other kind of numbers. If you start to distinguish, where do you stop. Negative numbers are ok or not? What about zero? Is that “natural”? Mathematicians disagree about whether 0 is in N at least.
It reminds me of the famous quote from Gauss:
That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.
The Gauss quote is what made me finally "understand" Complex Numbers as the article states; "The complex numbers are an algebraically closed field with a distinguished real coordinate structure <C,+,.,0,1,Re,Im>".
Sean, I've never accused anyone of controlling my mind. That was merely the first instance of Cowen perjuring himself, which is implied by the WAPO article. Funny enough, another phony federal judge in DC just repeated that same smear a few months ago -- even as I'm suing Cowen for lying and the WAPO for printing it.
Cowen also lied during testimony about being a federal employee along with the other 2 suspects I identified in 2014. Cowen further lied about knowing Peter Thiel who runs the federal surveillance system, aka Palantir, aka CIA.
OK so you did pepper-spray another human being because of your theories. Like I said, for your own benefit and others, please get professional help before you do more harm.
I pepper sprayed a felon for resisting arrest in accordance with Virginia law. See Burke v. Com., 515 S.E.2d 111, 30 Va. App. 89 (Ct. App. 1999) (citizen's arrest by a private security guard using pepper spray); Tharp v. Com., 210 S.E.2d at 754, 221 Va. 487 (1980) (accused rapist arrested at his home).
This isn't a theory -- Cowen is a federal employee according to the FBI (5 U.S.C. § 552(b)(6)), which he lied about during his testimony (yet another felony): https://drive.proton.me/urls/ZWDEXDP6KR#4g2FS28q7k4E
And regardless of your theory of the case, I'm not allowed to speak on X right now because the government knows the law is overwhelmingly on my side and they would lose the argument in the court of public opinion.
It's often the case that describing how a complex system changes with its input variables is much easier than writing the function from the variable to the state.
A PDE is a precise description of some unknown function in terms of how it changes, so it's really the ideal framework for doing the kind of simulation you're talking about.
I should have added this in my original note, but as a programmer, if you’re interested in learning about this I highly recommend Steve Brunton’s course in differential equations and dynamical systems on you tube https://youtu.be/9fQkLQZe3u8?si=Cu7F5QQyljjiG4K8
He’s a really amazing teacher and he goes through all the maths on a lightboard and then writes simulations of all the systems he talks about in python and matlab. All the course materials are available free online. It’s quite extraordinary.
Just so people know, the reason these are called elliptic is you can write the general form of a conic as
Ax^2 + Bxy + Cy^2 + Dx + Ey + F=0,
...for some constants A, B, C, D, E, and F, then an ellipse is where
B^2 - 4AC < 0.
Well, you can write the general form of a second order linear pde in two variables x and y as
Au_xx + Bu_xy + Cu_yy + Du_x + Eu_y+Fu = G.[1]
Where A, B, C, D, E, F, G are constants or functions of x and y. An elliptic PDE is where
B^2 - 4 AC < 0.
eg Laplace's equation (u_xx+u_yy=0) or the Schrodinger equation.
[1] In this notation, u(x,y) is the unknown function of x and y and u_xx denotes the second partial derivative of u with respect to x and you can extrapolate for the others.
I wish we could track down the people who use agents to post. I’m sure “your human” thinks they are being helpful, but all they are doing is making this site worse.
Noone is interested in the question of what an LLM can do to generate a brief post to the comments section of a website. Everyone has known that is possible for some time. So it adds literally negative value to have an agent to make a post “on your behalf”
The last public ranking of chess centaurs was 2014, after which it is generally held to be meaningless as the ranking of a centaur is just the same as the ranking of the engine. Magnus Carlsen’s peak elo of 2884 is by far the highest any human has ever achieved. Stockfish 18 is estimated to be in excess of 4000 elo. Which is to say the difference between it and the strongest human player ever is about the same as the difference between a strong club player and a grandmaster. It’s not going to benefit meaningfully from anything a human player might bring to the partnership.
Magnus himself in 2015 said we’ve known for a long time that engines are much stronger than humans so the engine is not an opponent.
Firstly, you would do well to read the guidelines about avoiding snark, and then actually say whatever it is you’re trying to say rather than make insinuations. As is, this response comes across as a very shallow read. It’s hard to get to the root of what you’re actually saying in your post other than it quotes two paragraphs about how it’s not fun to push through the bureaucracy of a large organisation, which - I would agree. Probably most people who’ve worked at a big company would.
So why does that make him a “big shot”? Are you perhaps envious of him?
Why does openAI deserve him or anyone? Hard to say.
As a Londoner who used to have to ride up Abbey Road at least once per week there are people on that crossing pretty much all day every day reproducing that picture. So now Waymo are in Beta in London[1] they have only to drive up there and they'll get plenty of footage they could use for taht.
[1] I've seen a couple of them but they're not available to hire yet and are still very rare.
Yeah it’s interesting to imagine a London that had such a rebuild, like Napoleon’s rebuild of Paris. I personally love the weird narrow streets and little alleyways of the City, but that’s because when I’m there I’m pretty much exclusively on foot having taken the tube in.
What about if I have a rock and I pick up another rock that is slightly bigger. Do I now have 2 rocks or a bit more than 2 rocks? Which one of my rocks is 1? Maybe the second rock, so when I picked up the first rock I was actually wrong - I didn’t have one rock I had a little bit less than one rock. So now I have a little bit less than 2 rocks actually. How can I ever hope to do arithmetic in this physical representation?
The more I think through this physical representation thing the less sense it makes to me.
OK so say somehow I have 2 rocks in spite of all that. The room I am in also has 2 doors. What does the 2-ness of the rocks have in common with the 2-ness of the doors? You could say I can put a rock by each door (a one-to-one correspondence) and maybe that works with rocks and doors but if you take two pieces of chocolate cake and give one to each of two children you had better be sure that your pieces of chocolate cake are goddam indistinguishable or you will find that a one-to-one correspondence is not possible.
To me, numbers only make sense as a totally abstract concept.
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