I really love that video.
The space of possible abstractions for any given phenomenon is vast, yet we almost always just assume that real numbers will do the trick and then begrudgingly allow complex ones when that doesn't work. If we're not lucky we end up with the wrong tool for the job, and we haven't equipped people to continue the exploration. It's a bias with some pretty serious consequences (thanks... Newton?).
I don't think I've seen the inadequacy of number-systems-you've-heard-of demonstrated so clearly as it is done here.
Well, don't leave us hanging! What are some of your favorite hot games on top of switches?
"Example 1: Numerical Context
Let's consider two real numbers, a and b. If a is neither greater than nor less than b, but they aren't explicitly equal, the relationship is ≹"
How can that be possible?
But the numerical context can still be correct: imaginary numbers for example don’t have such a property.
Or more generally, vectors. They don't have a total order, because as we define "less than"/"greater than" in terms of magnitude (length), this means for any vector V (other than 0), there's an infinitely many vectors that are not equal to V, but whose length is equal to length of V.
Is this is what ≹ is talking about?
Note that Games do _not_ form a field: there is no general multiplication operation between arbitrary games.
Imagine you have 2 irrational numbers, and for some a priori reason you know they cannot be equal. You write a computer program to calculate them to arbitrary precision, but no matter how many digits you generate they are identical to that approximation. You know that there must be some point at which they diverge, with one being larger than the other, but you cannot determine when or by how much.
The 1/3 * 3 argument, I found the most intuitive.
Thats what's counter intuitive to people, it's not an issue with 1/3. That has just one way to write it as decimals, 0.333...
All the decimals that recur are fractions with a denominator of 9.
E.g. 0.1111.... is 1/9
0.7777.... is 7/9
It therefore stands to reason that 0.99999.... is 9/9, which is 1
Let x = 0.99...
Then 10*x = 9.99...
And if we subtract x from both sides, we get:
10x - x = 9.99... - x
And since we already defined x=0.99... when we subtract it from 9.99..., we get
9x = 9
So we can finally divide both sides by 9:
x = 1
The geometric series proof is less fun but more straightforward.
As a fun side note, the geometric series proof will also tell you that the sum of every nonnegative power of 2 works out to -1, and this is in fact how we represent -1 in computers.
Isn't the sum of any infinite series of positive numbers infinity?
\2 is "not always" ..
Consider SumOf 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ...
an infinite sequence of continuously decreasing numbers, the more you add the smaller the quantity added becomes.
It appears to approach but never reach some finite limit.
Unless, of course, by "Number" you mean "whole integer" | counting number, etc.
It's important to nail down those definitions.
The same argument I mentioned above, that subtracting 0.99999... from 1 will give you a number that is equal to zero, will also tell you that binary ...11111 or decimal ...999999 is equal to negative one. If you add one to the value, you will get a number that is equal to zero.
You might object that there is an infinite carry bit, but in that case you should also object that there is an infinitesimal residual when you subtract 0.9999... from 1.
It works for everything, not just -1. The infinite bit pattern ...(01)010101 is, according to the geometric series formula, equal to -1/3 [1 + 4 + 16 + 64 + ... = 1 / (1-4)]. What happens if you multiply it by 3?
...0101010101
x 11
-------------------
...0101010101
+ ...01010101010
-------------------
...11111111111
And decimal "...999999" is an infinity, which should immediately set off red flags and tell you that you need to be extra careful when analyzing it.
In computers your series of 1s is not infinite, there's a modulus that steps in. And this analysis depends on the modulus being an exact power of the base. But you could make a system that's decimal but has a modulus of 999853, for example, and then "-1" would be 999852.
That isn't quite correct. The series of 1s really is conceptually infinite. That's why we have sign extension. The analysis (of the sum of all natural powers of 2) will work for any modulus that is an integral power of 2, including a modulus where the integer to which 2 is raised is infinitely large. Such an infinite modulus will still be evenly divided by a finite power of 2 -- as well as by itself -- and so it will disappear whenever you're working in any finite modulus that is a power of 2 -- or when you are working modulo 2^ℕ. The modulus of 2^ℕ will prevent any distinct finite integers from falling into the same equivalence class.
This is what enables you to have an infinite series of leading 1s, or leading patterns, without problems.
You can 'represent' the process of summing an infinite number of positive powers of x as a formula. That formula corresponds 1:1 to the process only for -1 < x < 1. However, when you plug 2 into that formula you essentially jump past the discontinuity at x = 1 and land on a finite value of -1. This 'makes sense' and is useful in certain applications.
There is some weird appeal to the Zeta function which implies this result and apparently even has some use in String Theory, but I cannot say I was ever convinced. I then dropped the class. (Not the only thing that I couldn't wrap my head around, though.)
Do that multiplication and you'll find the result is (1 - 2x + 3x² - 4x⁴ + ...). So the sum of the sequence of coefficients {1, -2, 3, -4, ...} is taken to be the square of the sum of the sequence {1, -1, 1, -1, ...} (because the polynomial associated with the first sequence is the square of the polynomial associated with the second sequence), and the sum of the all-positive sequence {1, 2, 3, 4, ...} is calculated by a simpler algebraic relationship to the half-negative sequence {1, -2, 3, -4, ...}.
The zeta function is just a piece of evidence that the derivation of the value is correct in a sense - at the point where the zeta function would be defined by the infinite sum 1 + 2 + 3 + ..., to the extent that it is possible to assign a value to the zeta function at that point, the value must be -1/12.
https://www.youtube.com/watch?v=jcKRGpMiVTw is a youtube video (Mathologer) which goes over this material fairly carefully.
Edit: Not in many programming languages. In IEEE-754 inf == inf. In SymPy too oo == oo, although it's a bit controversial. Feels sketchy.
Contrived, but only thing I could think of.
Now if you really think about, a number of a given magnitude on x axis also isn't exactly "equal" to a name of same magnitude on y axis or vice versa. Other wise, -5 and 5 should be equal, because they're the same magnitude from 0.
Edit: oh, I see what you mean. 1 is not larger or smaller than i, but it also doesn't equal i.
I've never really seen this notation used, but it could have some use in partially-ordered sets.
(a+b)^2 != a^2 + b^2
Strictly you should write something like
¬[∀ a,b (a+b)^2 != a^2 + b^2]
The question is why the page says "imagine two real numbers that aren't comparable".
\inf and $\inf + 1$ comes to mind but I don't think it really counts
That just depends on the numeric structure you're working with. In the extended reals, +inf is equal to +inf + 1.
In a structure with more infinite values than that, it would generally be less. But they wouldn't be incomparable; nothing says "comparable values" quite like the pair "x" and "x + 1".
You could imagine two fuzzy numbers with the same 'crisp' number having different membership profiles, and thus not being "equal", while at the same time being definitely not less and not greater at the same time.
Having said that, this all depends on appropriate definitions for all those concepts. You could argue that having the same 'crisp' representation would make them 'equal' but not 'equivalent', if that was the definition you chose. So a lot of this comes down to how you define equality / comparisons in whichever domain you're dealing with.
> To conclude, the ≹ symbol plays a crucial role in providing a middle ground between the traditional relational operators.
As a PhD student in math, I have never seen it before. I do not believe that it plays any crucial role.
This symbol for it may be useful, but it's the concept that matters.
The author of the original article uses it correctly - think about it more in regards to importance for their example.
The business is no more or less important than the developer, but they are NOT equal.
It doesn't have to mean importance though, just the method by which you are comparing things.
Monday ≹ Wednesday
Come to think of it, it should be called the 'No better than' operator.
That’s only true for a total order; there are many interesting orders that do not have this property.
It holds for the usual ordering on N, Z, Q and R, but it doesn’t hold for more general partially ordered sets.
In general one has to prove that an order is total, and this is frequently non-trivial: Cantor-Schröder-Bernstein can be seen as a proof that the cardinal numbers have a total order.
Not in a partial order.
For example in this simple lattice structure, where lines mark that their top end in greater than their bottom end:
11
/ \
01 10
\ /
00
You can actually see this kind of structure everyday: unix file permissions for example. Given a user and a file, the permissions of the user are is an element of a lattice where the top element is rwx (or 111 in binary, or 7 in decimal, which means the user has all three permissions to read, write, and execute) and the bottom element is --- (or 000, in binary, or 0 in decimal, which means the user has no permissions). All other combination of r, w, and x are possible, but not always comparable: r-x is not greater nor lesser than rw- in the permissions lattice, it's just different.
For example, set inclusion. Two different sets can be neither greater than not smaller than each other. Sets ordered by inclusion form a partially ordered lattice.
So you can end up with a sequence "d > b > a" and "d > c > a", but "c ≹ b".
Defining how tie-breaking for those cases are deterministically performed is a big part of the problem that CRDTs solve.
Such a thing is called a partial ordering and a set of values with a partial ordering is called a partially ordered set or poset (pronounced Poe-set) for short.
It really is an interesting thing. In fact, as human beings who by nature think in terms of abstract, non-concrete units (as opposed to mathematically precise units like a computer program), we tend to compare two related things. They might belong to the same category of things, but they might not be eligible for direct comparison at all.
Once you internalize partial ordering, our brain gets a little more comfortable handling similar, yet incomparable analogies.