Subnostr

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.

YODL 6mo ago

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).

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ᴛʜᴇ ᴅᴇᴀᴛʜ ᴏꜰ ᴍʟᴇᴋᴜ 6mo ago

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.

YODL 6mo ago

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.

ᴛʜᴇ ᴅᴇᴀᴛʜ ᴏꜰ ᴍʟᴇᴋᴜ 6mo ago

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.

YODL 6mo ago

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.

ᴛʜᴇ ᴅᴇᴀᴛʜ ᴏꜰ ᴍʟᴇᴋᴜ 6mo ago

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.

YODL 6mo ago

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.

ᴛʜᴇ ᴅᴇᴀᴛʜ ᴏꜰ ᴍʟᴇᴋᴜ 6mo ago

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.

YODL 6mo ago

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.

ᴛʜᴇ ᴅᴇᴀᴛʜ ᴏꜰ ᴍʟᴇᴋᴜ 6mo ago

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.

YODL 6mo ago

Follow most of your line of reasoning. Agree to disagree slightly 🤷‍♂️

ᴛʜᴇ ᴅᴇᴀᴛʜ ᴏꜰ ᴍʟᴇᴋᴜ 6mo ago

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.

ᴛʜᴇ ᴅᴇᴀᴛʜ ᴏꜰ ᴍʟᴇᴋᴜ 6mo ago

and yes, that's the thing, you have to have a thing before you can talk about a thing.

YODL 6mo ago

First axiom of set theory is "there exists a set" lol. It's sometimes left out as it's sort of philosophical.

ᴛʜᴇ ᴅᴇᴀᴛʜ ᴏꜰ ᴍʟᴇᴋᴜ 6mo ago

probably that means philosophy precedes topology, which then grows into sets, and then numbers and geometry.

YODL 6mo ago

I dunno. I'm making coffee 😮‍💨