Yes, and?
Discussion
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
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.
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
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
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, 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...
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
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?
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.
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.
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.
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).
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.
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.
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.
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.
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.
Follow most of your line of reasoning. Agree to disagree slightly 🤷♂️
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.
and yes, that's the thing, you have to have a thing before you can talk about a thing.