Replies (71)
I can't say I've ever given that much thought. I'll just go with whatever you think🙏🏻.
It's not hard to follow, if you just reason it through. Maybe the tricky part is realizing there are finitely many 60 letter or less combinations of letters, a subset of which are meaningful and define a positive integer (rest are gibberish as far as we're concerned). This can only refer to a finite set of numbers, so there are ones not defined. The least of these satisfies above short sentence. Contradiction.
Its a novel twist on a typical flavor of paradox, to me anyway. Neat
My brain isn't firing on all cylinders today so I'm just defering to your (Almost certainly correct) explanation😎.
I did see you posting a lot yesterday lol. Glad you're still around in force from time to time
letters aren't digits
Yes, and?
Saw that too
I know i chat shit from time to time but I'm glad that some people do appreciate my random outbursts and my niche musical offerings😬😂.
Yeah, I hope to at least be able to find folks on here for a long time to come
Being a little snarky since I'm fairly certain you missed something in the post. Mostly a copy.paste from Wikipedia so not likely I am wrong
Letters? Shouldn’t it be numbers?
You're second person to make this point... maybe it's more confusing than I thought.
You can write statements with letters that define numbers, eg "the smallest number greater than 3" defines 4 (in integers).
Take all 60 letter or shorter statements (finitely many). Toss out those that are gibberish or don't define a integer as they're irrelevant, and you have a (even smaller, though besides the point) finite set of sentences, hence integers. That leaves infinitely many not defined by such statements. There is a smallest positive one. Bingo bongo
I get all that. But wouldn’t it make more sense with numbers?
It's kind of a twist on saying things about statements themselves, leads to contradictions. The classic "this statement is false" type of thing, or "the set of all sets that aren't a memeber of themselves" imo
Takes less letters to write one trillion than fifteen billions for example.
yeah, i read the thing, it's still retarded. it's like saying there's a finite number of positive integers that can be written with 10 digits. yeah, no kidding.
what is the ratio of any finite number to infinity? the identity!
some of these abstractions in mathematics around infinity are comical in my opinion.
but then, imaginary numbers are quite hilarious, too. i had some model in my mind at one time as i pondered on the matter of the square root of -1 and i forget what i came up with. ah yes, it's the inversion of the rule about signs. negative numbers themselves are imaginary. they literally are like not-numbers. the numbers that are not there. i think that led me to some idea that why they are useful is because they have a reverse temporal property, similar to how multiplication and division are temporal opposites, and why division is the most expensive basic arithmetic operation, can't be optimized very much, always has to be iteratively computed based on the number of places in your representation, so it's ... well, a constant time operation.
Correct, but as each is defined here and in under 60 letters, they are excluded from consideration. You're not constrained to writing out the way we say the number, you can use all forms of description.
What do you mean by the second sentence? Like writing ten times ten instead of one hundred ?
base32 uses 26 letters and 6 digits
you can consider a byte to be base 256 also, variable integers that use a continuation bit are like a compact base 128 encoding
i mean, keep in mind, you can represent 8 bits as a symbol with a grid of 2x4, that's really not that complicated a set of symbols. also there will be a natural set of symmetries to them, inverses, and mirrors and rotations
It's not retarded. That aside I'll comment on complex numbers.
They are interesting from a few different perspectives. One is as THE (up to isomorphism) algebraic closure of the real numbers - which means all polynomials have roots in it. Sort of the final extension you get when doing stuff with inifinite fields and the "standard" polynomial stuff.
Not sure I followed the multi/division being temporal opposites stuff, but I know what you mean in terms of compute time. Don't see connection with that to negative numbers, which are really just subtraction, the opposite of addition, but no more time consuming to compute
Yeah
I don’t know about this. Never saw a maths theorem with letters before. Maybe it’s something for fun
Letters instead of digits. But you get my point
Yes, but saying something like "the least number not expressable with fewer than 3 base 10 digits" (1000) is not an interesting framework. This is meant to draw attention to how loose treatment of what feels like rigorous definitions leads to problems.
Wish we had voice rn lol. You get the gist. Highlights how even formalized language can lead to basic contradictions if you're not more careful. Came up when reading some stuff today on foundations of math... you'll love my upcoming blog post if you like this. Soontm
Ignore that I messed up my example slightly with 1000 instead of "100" above. Just reread it
yeah, sloppy definitions is why i hate dynamic typed languages. i have enough trouble with getting sequence and signs in a muddle with my dyslexia to then pile on top of it ambiguity.
Yeah, it was these types of issues that spurred efforts at formalizing logic in last century, my understanding. Language like this, when not more carefully built up, can do all sorts of self referential tricks.
Even when you do make the efforts to make it concrete, you gave issues as Gödel famously showed a bit later. Just kinda cool to trip out on imo
yes, but my point is, that negative numbers are an abstraction. you can count 3 apples, but you can't count -3 apples because they are not in front of you to count. they are hypothetical negatives, living in some ghost world.
negative *number* what does number mean? it means to count, and counting means addition, 1 and 1 and 1 and 1...
it's not that they have no use because they "don't exist" neither do any kind of numbers. real numbers are just as bad, everything seems to indicate that there is a finite precision to the substance of this universe, which is the opposite of what a "real" number is. "real" should be the integers, because this highlights the fact that there is no such thing as perfect in this universe, only approximations with varying levels of precision.
negatives definitely are the opposite of addition though. you add one, you subtract one, and you are back to zero. so subtracting 1 is the temporal negative of adding 1.
same as multiplying, they are opposites in one way, and the same in another, as scalars, they are the same, as vectors, they are opposite.
anyway, nothing stopping anyone from making an arbitrary number of symbols and making a number base out of them based on their sequence in a place of representation. it's just not really the same as a number, if you don't specify base, which is implied by representation.
You sound like a constructivist 🤭
Which is to be fair a school of thought, just far less popular.
All numbers are abstractions, but get your point. Zero used to blow peoples' minds too though
yeah, paradoxes are real enough though. even though they by definition don't make sense. but sense itself couldn't exist alone. it has to have a spooky slanted upside down inside out back to front, and reverse form.
like what i was saying about imaginary numbers. what exactly says that multiplying negative numbers should lead to an inversion of the sign? that would imply that it's actually a number itself, like one and zero, and the multiplication is *addition* but when you multiply *negative* numbers you XOR the "sign" bit. which is how it's implemented, also. why does it have to be XOR on the sign? it could be AND or OR, so you can only get a negative answer if both are negative, or that any negative and both are negative, instead of the flip-flop of repeated multiplication with a negative does.
my point being that it's not a scale, it's a vector, and the vector is time and space.
the fact that these spooks have utility tells you that the vectors relate to the "sense" of things as much as the nonsense.
makin me open notedeck to keep up with your gd fast typing.
you're confounding slightly, i think, expressing numbers in a system (whatever base you choose), vs defining things with a more general language. At least that seems to be one of the issues. I fully understand how you can do larger numbers with fewer slots in a really high-base system, that's clear. This sort of thing comes up when you try to closely analyze "propositions" in math, and I'm not equipped to really go any deeper, just a tourist still.
yeah, zero drives me crazy. in computers, zero stands in for first, so you use one encoding, and have two semantics, cardinal (counting) and ordinal (sequence). this is why off-by-one bugs are very common in iterator code.
the fact that computers encode everything as binary also gets quite confusing because of the way that AND/OR/NOT/NOR/XOR operations can happen, but they make changes in the values that don't comport to "normal" arithmetic operators. and multiply and divide themselves are essentially adding a dimension on top of their base.
idk...
anyway, that's the thing about arithmetic. it is about space and time, and counting. that's where the "met-" part comes from in the name because it's based on counting. and rhythm too, which is also about counting, and time.
yeah, i just have a very visual brain. when you use these words i get pictures of processes, and i naturally see scales and vectors, symmetries and inversions.
a lot of what seems absurd is just a reflection of what is sane.
random number generators, for example. people like to split hairs about whether it's really random if you can model it, but taht's teh thing, you can't model it until after it's happened. it's because the main mechanism of PRNGs relates to rounding errors and overflows being sent around in a circle, asymmetric cryptography is entirely built on top of the entropy caused by clockwork arithmetic, which in practise just means that bits fall off the left and they are gone, but because of the permutations and the rules of arithmetic, you don't get zero straight away, you get less... something... which we call entropy. to make it secure, you need good entropy, and then it's hard to figure out what was lost because it was more complicated than the thing it was added to.
haha. anyway. i love arithmetic. i first fell in love with it via the mandelbrot set but cryptography is even cooler. and the related coding systems, the principles of representation, integers, dimensions...
Don't get me started on off by one. I'll never get indexing straight! lol
But obviously everything should start indexing at zero!
The constructivist (i think that's the term, i just learned it) school of philosophy/math, doesn't see any proposition or its negation as necessarily the only possibilities. This is unusual since all the math and reasonsing we do typically assumes this implicitly. There's a term "law of excluded middle" that is related, you can sorta guess what it means.
In this vein, there were issues with infinity/real numbers (infinite sequences lead to irrational numbers) a 100-200 years ago, which went glossed over for a long time, but finally had to be tackled, and that's partly what led to for example the Dedekind construction of the real numbers. It lays out a concrete canonical definition of what they are set-theoretically. You might enjoy a look into that.
Still have small issues with your framing of crypto, but can't get into it now. Another time!
Oh, and one other thing I can't let go of, I think you once described finite fields inaccurately. They're entirely classsified, and all of order p^n for any n>=1
haha, infinite sets, there's a perfect example of a paradox.
but regarding counting/sequence, when that distinction became clear to me i started to notice it all through language, i mean, where the semantics of words relates to sequence, or number (or size) there has to be some sense to it.
like, really, it should not be chapter 1, you call it the first chapter. first can be represented by 1 but it wouldn't be first if there wasn't a state before it. so that's why they invented zero. because of that implicit contradiciton between count and sequence. it wouldn't make sense to say that the ordinals are like halves, they are integral in the same way, but first has to be nothing, before you can start counting anything. nothing is infinite, also, until it starts, then you have finity...
now that's one i've not thought through very much. positive infinity of counting. just because it starts at zero, doesn't mean it has to end. i'd argue that implies it can't.
i may have been confusing them with arithmetic groups, which are a consequence of finite fields.
i just think of finite fields like the old school games where you go across the edge on the right and come back around on the left. the rules about what happens to you in that process though, that's where the devil is. does it wrap around? in what way, let's say it's binary, and we sequence the operation as one pass across the bits, then you can say "yes, this divide threw all the bits to the right one, so we can bring a new bit back from the left now" or if you add two numbers, and it overflows, how do you decide what way those overflowed bits go around the circle? is it a circle, or is it a mirror thing, so if 1010 overflows does it become 0101 on teh other side?
There are very good reasons to accept inifinite types of constructions. All of calculus is basically inifinite limits. Thats from what I've read, to the uncomfortable but necessary acceptance of formal infinity.
A simple example I like is Zeno's paradox. You take one step of size 1 on first second, half step in next half second etc. This is just breaking apart the act of two full steps done in two seconds, yet we can describe it as an infinite sequence of smaller and smaller steps (1, 1/2, 1/4 adds to 2).
Hmm. This is interesting. I take it that the definition is very precise. By that we don't mean the smallest number after which there is no way to define any larger number in 60 letters. We literally mean some oddball number that is "indescribable." It is probably prime or nearly prime and isn't close to any numbers with a compressible factorization.
But we just described it with the negative statement. This is kind of like Reyo's number but lacks the restriction to "first-order set theory."
What you are going to end up with is the set of all numbers that cannot be described in less then 60 characters, each of which cannot be the answer. So it ends up oscillating between the first two members of the set.
Sounds like U.S. political parties.
Need to reread this when not walking.
But yes, it IS interesting! Someone appreciates it
Will have to look up reyos thing you mentioned, but yeah, it's a flavor of the usual type of trap, using a second order type of logic expressed in what was assumed to be first order...maybe something like that.
yeah, i have in my mind a model like that as the driving mechanism of the motion of all matter in the universe, an infinite subdivision process.
I've found a couple more versions of the original paradox. There's another named after Richard. The two apparently are classified as definability paradoxes, as opposed to set theoretical ones like those of Cantor or Russel.
Will need to follow up on it tomorrow.
i've long been of the opinion that in the phylogeny of mathematics, first there is topology, then geometry, then sets. numbers are a a child of sets. geometry is based on surfaces and graphs, ie topology. sets are based on geometric relations, oppositions, adjacency, dimension (scale).
definitions probably fit within the topology/geometry section of the map, since they are about discovering the relations between concepts and things, which are basically vectors and scalars.
I could write a short essay in response...
TLDR is if put sets first, since that stuff is foundational to the rest. Topology has a formal math definition which may surprise you, not exactly what you'd expect, just a few rules about sets inside a larger space, and continuous functions types of things.
i think that topology leads to sets, by the categorisation, but sets don't exist until you have a variety of elements with uniqueness first. and geometry goes second because you use geometry to build the phylogeny of sets. so maybe topology -> sets -> geometry
like, you can't have geometry without polyhedra, or vectors, but you can have topology without sets. a circle is not a set, it is an element of a set. same can be said of a plane or a bounded surface. they form into a set of sets, but they precede the existence of sets.
so i still say that topology is the root. topology delineates the form from the mold, the day from the night.
We should define this more formally. Twenty-six letters case-insensitive plus space, commas, and periods. Then we should start from 0 and find a many example as we can.
"Smallest natural number that cannot be defined by n characters [a-z ,.]"
0:1
1:1
2:1
3:3
4:3
Ah, I see how you're viewing things somewhat. Maybe based on intuitive notions of topology that's true, and historically that may be how things evolved. Set theory is now considered the foundational math, though it wasn't formalized until "modern" times when troubles began to arise in foundations of the existing math (as they sorta pushed its limits I guess).
Now the view is you start with set theory, build out some concepts and definitions, and other stuff is more special areas of interest within that framework.
Look up how topology is defined axiomatically and you'll likely be surprised at how different it is from what you would expect. Just a collection of sets and a couple rules about intersections and unions, that's it. Similarly with algebra. It's all sort of a modern movement from late 1800s onward toward more formalization and abstraction. Type of stuff I'm reading about now.
Not following the bottom line, but sure. I think getting worried over the details misses the point. I think you'd like this version better:
Let A be the set of all positive integers that can be defined in under 100 words. Since there are only finitely many of these, there must be a smallest positive integer n that does not belong to A. But haven't I just defined n in under 100 words?
From here, where you can read a bit more
ok, maybe what's different in my mind is that i am thinking about it as a phylogeny, that is, you have to look at what is primary before you can get to a given domain. set theory is founded on the fundamental topological concept of a manifold. you can't have a set without that first existing manifold.
imagine the universe just started. what is the first thing that is going to happen? space would be divided up. the shapes would evolve into more complex patterns, this is all topology.
set theory is after the fact. topology lets you start from really zero. then you divide it, and you start to see the beginnings of phyla of things, which you have to have before you can start talking about categorising, grouping, comparing and dividing them from each other.
yes, that's the key, dividing. dividing space, be it a surface or a line or a volume, is the fundamental basis of topology. the ways in which you do this form the first sets.
i still say topology is the root of all mathematics.
No, I understand the paradox. I just want to know the answer once you remove the paradox. If I change 100 to 10 then the paradox goes away and there is some number that meets the criteria. I want to know what it is!
isn't it just the finite field represented by the set of symbols and places?
No. You can easily define numbers larger than the set. Which means the largest number you can describe with words is somewhere within the set.
you can only define numbers larger than the set if you allow the set to be expanded in some way. more symbols, more places.
Sorry I got the backwards. "Smallest number you can't describe with words"
We're gonna have to disagree out the gate. A manifold has a precise definition as a surface which is locally Reimanian, something like that. I'd classify it as one of the more elaborate constructs, and less primitive or whatever than a simple set. Afterall, you need set elements to talk about before you can define a manifold. All the formal stuff I've seen looks like "a group is a SET, with the following additional properties...".
Reading the rest I do see how you mean though, so I'll stop getting hung up on technical definitions. Yeah, I'd think patterns or differences come first, then the notion of distinct things. That's how I'm roughly understanding you.
zero? nothing? what is that but the purest form of placeholder?
i guess the thing i'm disagreeing on relates to a definition of a surface that requires discrete or compositional elements to describe.
a point is before a set, a set is a collection of points. a line can be divided, then you can make a set, a set could be said to be all sets of points in a single dimension. you can make that dimension infinite, or you can make it circular, again, the geometry precedes the establishment of a set.
a point is the primary unit of topology. so i guess i'm splitting hairs.
i just don't agree with the post-hoc nature of sets, being primordial compared to the ad-hoc nature of divisions of surfaces.
If you change the letter limit to 10, what is the answer indeed... I'll have to let you elaborate, if you feel moved to, as I have no idea
and yes, that's the thing, you have to have a thing before you can talk about a thing.
Follow most of your line of reasoning. Agree to disagree slightly 🤷♂️
First axiom of set theory is "there exists a set" lol. It's sometimes left out as it's sort of philosophical.
yeah, correcting myself. the topology precedes the set.
a circle, as in a theoretical perfect circle in a continuous (real) field is not an infinite set of points. that is an approximation of a circle, just as you can't get a precise number, in any number base, for the ratio of circumference to radius.
probably that means philosophy precedes topology, which then grows into sets, and then numbers and geometry.
I dunno. I'm making coffee 😮💨
The symbols only have meaning if you give them meaning. Let's say I give you five places 0-9 and the lower case alphabet. So you write 'zzzzz' as the largest integer therefore '100000' must be the smallest integer that can't be defined right? Well I'd argue that 'zzzzz' doesn't define anything since it doesn't define its own base. What if you had written 'tree3'? Is that a base 36 or the usual huge number we normally define it as?
I have no idea either. That is why I stopped a 4 instead of ten. I am not even 100% on that one. There are a lot of 4 character strings.
