HexesofVexes

Thanks for the bedtime reading!

I mostly deal with foundations of analysis, so this could be handy.

Ehh, completed infinities give me wind…

Reals are just point cores of dressed Cauchy sequences of naturals (think of it as a continually constructed set of narrowing intervals “homing in” on the real being constructed). The intervals shrink at the same rate generally.

1!=0.999 iff we can find an n, such that the intervals no longer overlap at that n. This would imply a layer of absolute infinite thinness has to exist, and so we have reached a contradiction as it would have to have a width smaller than every positive real (there is no smallest real >0).

Therefore 0.999…=1.

However, we can argue that 1 is not identity to 0.999… quite easily as they are not the same thing.

This does argue that this only works in an extensional setting (which is the norm for most mathematics).

Wow… They want to give AI even more mental illness and crippling imposter syndrome to make it an expert in one niche field?

Sounds like primary school drop-out level thinking to me.

Capitalists AND Reviewer 2… Never underestimate the power of Reviewer 2 in publication!

“How do we stop the world’s smartest people from realising what we’re doing?”

“Let’s make them fight among themselves and call it a meritocracy; we’ll limit their funding and let them keep themselves busy with political infighting!”

And, as a mathematician who has been coding a library to create scaled geometric graphics for his paper, I hate -0.0.

Seriously, I run every number where sign determines action through a function I call “fix_zero” just because tiny tiny rounding errors pile up in floats, even is numpy.

So about 3150 pints of blood (10.5 being average for an adult).

Sounds doable XD

Edit: New ethical dilemma just dropped - kill 300 to forge the sword, or deny 3150 people blood in an emergency to forge the sword…

I have a rare medical condition that makes my coughs sound like “scihub” and “libgen” around undergrads.

I would like to investigate it further to seek a cure, but sadly the medical journals I’d need access to are paywalled. Oh well.

I’m arguing from the standpoint that we establish the idea of counting using the naturals - it’s countable if it maps to the naturals, thus the link. Apologies for the lack of clarity.

On the contrary - to be countabley infinite is generally assumed to mean there exists a 1-1 correspondence with N. Though, I freely admit that another set could be used if you assumed it more primitive.

N is the set of “counting numbers”.

When you count upwards you start from 1, and go up. However, when you count down you usually end on 0. Surely this means 0 satisfies the definition.

The natural numbers are derived, according to Brouwer, from our intuition of time of time by the way. From this notion, 0 is no strange idea since it marks the moment our intuition first begins ^_

Platonism Vs Intuitionism would like a word.

No, it was a philosophy PhD student on his 2nd bottle of tequila apparently.

“A number is how far along you are on your journey, infinity is the horizon you journey towards.” - Some guy I met in a pub

Responsible adults are boring and should live a little!

Yup, he definitely belongs in a stable.

Or is “strong and stable”?

So, let’s say you write down the words “fire is a chain reaction between carbon and oxygen that produces heat”. You’ve characterised fire yes, but is that sentence itself the fire?

Let’s say you write down the equation describing this reaction so you can play with it and manipulate it. Is this fire, or just a convenient way to talk about it?

I’d argue neither of these are fire, and both will never completely describe a fire (though they come damn close).

You know, I’d never considered that, but yes that does indeed seem to be the case!

See now, I’d argue that the language comes after the mathematics. For example, I walk to work each day; part of walking to work is trying to find the route that lets me lie in the longest.

Now, humans are pretty good at exploring and finding alternative routes between locations, and they also tend to locate the shortest route given enough time.

Trying to explain how this intuitive activity works necessitates the use of graph theory. The graph theory was something our brain had constructed in the background, but it wasn’t entirely conscious. Trying to explain this in natural language would take pages, however…

Given a set V of street intersections, and a set E of streets connecting two intersections, and a set W of weights assigned to each E. I can calculate the shortest route by applying one of the pathfinding algorithms (which are expressed in this notation).

This explanation will cover any pathfinding problem, but it’s not great at conveying what is a really happening. The language we must use gets in the way of conveying the mathematics that is going on.

We do need a language (telepathy not being on the menu), but that language is a separate entity from the mathematics itself.

There are “mathematical languages”, but these are present to describe mathematics. There are mathematical theories of language, but again the language itself is not mathematics - its structure, however, has mathematical properties.

I suppose you could say “fire has the property of being hot, but it isn’t hotness itself”? Language is used to communicate mathematics, but it is not mathematics itself.

Now, this is not to discount notational developments in easing communication - that’s a great branch as you have to check your new language and its rules match the mathematics it tries to describe. However, again, it’s important not to conflate the thing you are describing with the thing you are using to describe it!