5 ms·
In the real world we have atoms, but atoms are disturbances in the wave function of the universe, which may not be discrete. We know atoms pop in and out of ex
by Aicy 4y ago
In the real world we have atoms, but atoms are disturbances in the wave function of the universe, which may not be discrete.
We know atoms pop in and out of existence in matter anti-matter pairs. Maybe the universe is performing Banach Tarski under the hood, to make something from nothing?
> we use floating-point numbers, not reals, when doing actual calculations
I don't believe only calculations made by computers to be "actual calculations". Humans were calculating thousands of years before computers were invented, using integer or real numbers (e.g pi).
- moloch-hai 4y agoNo human or computer has ever used an actual real, transcendental number in any computation. It would take infinitely long.
- ttctciyf 4y agoI know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?
- Someone 4y agoI don’t think so. Programs that use numbers approximate e and π. It’s only programs that manipulate symbols that can use their exact values.
- gpm 4y agoAll programs manipulate symbols. All bits are is symbols. It's just sometimes we use those symbols to represent a subset of the integers (e..g. with the popular binary notation)... or a subset of the rationals (e.g. with the popular floating point notation)... or a subset of the reals that happens to include a transcendental number because we decided that some symbol (or combination of symbols) represents some particular transedental number.
- moloch-hai 4y agoThat is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.
- defrost 4y agoI absolutely agree with you from the outset . . . . . . and then I start thinking about that damn Bailey–Borwein–Plouffe formula. Although ( handwave ) all the transcendentalness is magically manipulated away leaving a simple computation for the immediate reveal of any arbitrary n-th (hexadecimal) digit of π But surely that's over simplification? Is this magic by wizards?
- moloch-hai 4y agoYou have an abstract definition of a sequence of digits, and an algorithm that reveals an actual digit from among them. It is surprising that you don't need to approximate an infinite series to get it, but everybody agrees that digit would show up there if you did one. You would still need to do an infinite amount of work to get the rest of them. It is magic of a kind by wizards of a kind, or anyway indistinguishable from it.
- defrost 4y agoJust to be clear, for any third party spectators, the surprise isn't that more work is required to get more digits .. the WTF moment for some is that no matter how large N is there's no need to do an increasing amount of work as N grows (there's no cost to "skipping to (not quite) the end"). It's another of those intuition challenging moments in math.
- moloch-hai 4y agoIt's the wackiest result I know of, and seems overwhelmingly more profound than BT.
- deleted 4y ago[deleted]
- CJefferson 4y agoBanach Tarski, in some sense, really can't make sense in the universe because the Universe is... discreteish, as you can't get smaller than the Planck Lenth (1.6 x 10^(-35)), and while that's very small, as far as maths is concerned that means the number of "identifiable points" in a sphere of any finite size is itself finite, not even a countable infinity. I realise that's all very dodgy, but of course we are also bringing in quantum mechanics -- but I consider it enough to say Banach Tarski doesn't apply :)
- lodi 4y agoMy understanding is that Planck Length is not a limit to the smallest possible thing; it's a limit to the most precise possible measurement. So there could in fact be lots of things happening at sub-Planck Length scales, but we could never observe them.