Irrationality of π

[terse palm]

Why do concious beings need to know/comprehend something fully for it to be true?

did you not see my point about max plancks constang

"constant

[ember marlin]

Then all real numbers are numbers, because all real numbers can be represented as a countable sequence of digits.

[gusty burrow]

1/3 need not be described as 0.3333...r, it depends on the base system you encode it in or expand into its decimal form so to speak. You can still represent anything that is with ratio as countable whole unit elements. ie 1/3 is the same as saying # relative to # # # so it has measurement and quantity.

[gusty burrow]

Digit sequences are not comprised of countable elemental units when they are in non-uniform flux.

You cannot represent anything irrational as countable elemental units and produce a measurement without making it rational.

[ember marlin]

YES. YOU. CAN. EVEN A NUMBER WITH INFINITELY MANY DIGITS STILL HAS COUNTABLY MANY DIGITS.

[gusty burrow]

I mean when irrationals play a role in the real world and effect quantity it is because they have landed and their effects are in rational form. The golden ratio is a good example of this.

[terse palm]

Just absolutely ignoring what I said and going into a new topic...

[gusty burrow]

No, it is like trying to count infinity or the total numbers that exist in an infinite sequence. It is a failure of measurement. You cannot measure what you cannot count. Irrationals literally mean unmeasurable in classical Greek.

[gusty burrow]

What? About Planck?

[terse palm]

Yes and about the fact that you dont need to comprehend something fully as a living being for it to be true

[ember marlin]

I. DON'T. GIVE. A FUCK. ABOUT. ETYMOLOGY. ESPECIALLY. NOT. FROM SOMEONE. WHO DENIES. THE EXISTENCE. OF COUNTABLE INFINITY.

[terse palm]

This thread is gonna drive everyone on mathcord clinically insane after some time

[gusty burrow]

"I am intrigued by your notion of “failure of measurement”, and I can somewhat see where you are coming from. But that does not mean that π is not a real number! You understand that 1/3 cannot be expressed in our number system, so why do you not agree with this same rationale for irrational numbers?"

Thank you for asking. The answer is well, because in some number system, any fraction or ratio can be expressed as a non recurring decimal when encoded in that format (which is a format of exponents of the base system lest not forget it holds no significance one over another). Irrationals cannot be expressed as a finite decimal point in any base system. And more important than decimals and base systems, is the inability to express an irrational as countable elemental units, it is in flux by definition. Just like an function with a limit that uses infinity in some form (aside from a few special examples). And positionally when encoded into any number system on a numberline, that flux is also non-uniform. There is no conceivable measurement, to measure one must be able to count units.

The square and its corner to corner with a line segment have being covered many times in the debate (can probably do a search for keywords and find all my mentions of it).

[gusty burrow]

Anyone with a working brain denies the existence of countable infinity. I think this debate is bad for you. Your responses keep getting more intense and now you are defending the ability to count infinity.

[terse palm]

So your problem is with irrationals not being a finite amount of information...? and im pretty sure no definition of irrationals says its "in flux"
see. my. point. about. plancks constant.

(Ik you already saw the message but youre still completely ignoring the idea that our knowledge of something isnt a requirement for it to exist)

[gusty burrow]

If you have a function, that uses infinity in the way that the irrationals do, or likewise a divergent function approaching infinity itself, that by definition is in flux. I view a function for an irrational with a limit that is the irrational, more similar to a function with a limit of infinity than a number. It cannot land by definition, it is in flux in any representation that involves numbers, number systems or countable elemental units. If we produce a measurement, it makes it rational.

Irrationals therefore cannot be operated on, and only ideological formalism with classical axiomatic abstractions as rationalizations/justifications permits such operations in my view.

[gusty burrow]

I do not see any problem with using plancks constant. Just like we use Pi or the Phi in approximation, we land it and make it rational.

[terse palm]

Tell me youre not reading what Im saying properly without telling me youre not reading what mi saying properly.
my main point was literally that plancks constant WAS rational so you should be perfectly fine with it, yet we didn't know it before 1900 but it still worked we just didn't know that it worked bc we didn't know it existed in the first place
read my original message PROPERLY for the proper explanation

[gusty burrow]

I have forgotten how Max Planck mathematically derived the e = hf, so I cannot really comment on that accurately. I would have to read up on my old notes again.

[terse palm]

why can it not land by definition?
take the limit to infinity of a sequence (6 + 1/n), it approaches 6, does that mean 6 doesn't exist? does that mean that 6 has no landing point?
also what do you mean "likewise a divergent function", A divergent sequence either diverges to +∞ or −∞ or is oscillatory, how is pi any of those...

[terse palm]

I literally have not learned anything about max plancks constant properly yet, all I had to do was look up "rational physics constant" to prove my point. thats how easy and intuitive it is.

I dont even remember the first digit of max plancks constant because of how briefly I went over it before I had enough information for my argument

[gusty burrow]

So how is it relevant? It is constant, it is rational, it is all good to me.

[terse palm]

Its relevant because we didnt always know it

yet it has always worked

[gusty burrow]

We didn't always know anything.

[terse palm]

exactly

yet everything has always worked

human knowledge is not a requirement for things to work

try saying that to yourself about pi please

[gusty burrow]

Pi does not exist in the physical world, infinitely perfect circles do not exist in the physical world or objects. Only in paths of motion if we assume a continuous distance and time metric and field of force, which produces other contentious issues in the groundwork of modern physics.

[terse palm]

Look to not get into unrelated details ill ask you about sqrt2 instead
does that too not exist in the physical world? cant you measure the diagonal of a square with side length one?

If you say no lines of length exactly 1 exist then great, by your logic 1 isnt a number either

[ember marlin]

Define "counting".

[terse palm]

I think this is another case of horrible use of math terms
she thinjs by countable infinity you mean a human can count all of it

but maybe im being unfair here surely shes seen stuff like N is countable but R isnt

[chrome cypress]

Yes, that square-diagonal analogy has very much been exhausted. I, too, a not a fan of repeated assertions. No need to say something twice.

Sapphire, you’ve got a good head on your shoulders when it comes to civil debate. I can tell when someone has put ample research and consideration into their writing. I also recognize mavericks when I seem them, since I feel that I relate to them more than standard convention. However, although many of the mathematicians who have made extraordinary discoveries also followed such unorthodox thinking, that does not mean that they were infallible. Bringing up your idea that the Greeks denied the existence of irrational numbers is an interesting key point, and it also ties into reason why it took almost 2000 years to prove the irrationality of pi. It takes continuous trial and error to reach a common notion, especially for those who must exert stronger efforts to reach mathematical conclusions.

If by “in flux” you mean that pi is not constant, that is simply not true. Just because all of the decimal digits in pi are inconceivable to our human comprehension does that mean that they do not exist (look towards other fields of study such as quantum physics and general chemistry!).

You can continue delving into this if you’d like, but you will receive vehement disagreement from almost any individual deeply ingrained in this study. I disagree with the mass antagonization you have received, and am rather interested in your claims; I think it’s great to quarrel (with complete respect from both parties), as it opens doors for extreme clarification and precision in prospected definitions. However, it might help to take into account that math is not just based on logical reasoning, it is also based on widespread consensus.

Perhaps you should take your findings onto some different platform(s) as well. Will esteemed individuals who have devoted their lives to mathematical degrees also agree with your justifications? I’m quite interested! Please don’t lose this curiosity of yours! As an artist, I can tell you this much: almost any famous artist, such as the revered Van Gogh, was hated in their time, and their efforts were only recognized many decades after death.

[ember marlin]

Why are you praising them for being obnoxiously wrong?

[gusty burrow]

No you cannot, it will always overlap or not reach with a line segment of any measurable length (made of equidistant countable elemental units). The sides of the square are 1, but no line segment can fit perfectly between the corners, I have covered this many many many times now. So I am not getting into a drawn out long explanation.

[ember marlin]

As an artist
So, not a mathematician and therefore a totally irrelevant perspective in this context.

[chrome cypress]

I’m praising them for sparking debate, of course. I like to see both sides of an argument before determining my position. I don’t find an issue with being wrong—the only issue I find is if somebody is wrong and they refuse to fix the given misconception.

[ember marlin]

Euclid disagrees with you.

[ember marlin]

This isn't a "debate", this is them being obnoxiously and stubbornly and publicly wrong.

[chrome cypress]

So the requirement to enter mathematical discussion is to have aptitude in such? Don’t see how that’s relevant.

[ember marlin]

if somebody is wrong and they refuse to fix the given misconception.
Yes. Yes, that is exactly what's happening.

[ember marlin]

No, the requirement to have something relevant to say about mathematics is to know mathematics.

[gusty burrow]

They are invoking something unrelated in the mathematical sense, so I assume they defer to something relevant to the discussion in my response. In actual mathematical terms "countable infinity" has nothing to do with counting it.

[ember marlin]

No, it absolutely does.

[gusty burrow]

He actually does not, have you read his work?

And not the formalist revision bs

[ember marlin]

I've read Postulate 2.

[chrome cypress]

I’m not here to argue with you. I’m here to give my advice to Miss. Sapphire. If there’s two things we should know about entering debate is that it is futile to argue with two types of people: people who refuse to admit lapses in thinking, and people who are unintelligent. I do not like to waste time, of course.

[ember marlin]

...are you insulting me? We have someone here who says irrational numbers change in value and you're insulting me?

[tired kettle]

Your philosophy seems to be technically self-consistent, though it's not one that I can buy into. It locks you out of many of the most beautiful areas of modern mathematics.

[chrome cypress]

I’m not. Please don’t take my words in the wrong way. I’ll admit that I’ve always been quite fond of you for your intelligence.

[ember marlin]

You said "I'm not here to argue with you," then you said "it's a waste of time to argue with stupid and stubborn people", and then you said "I don't like to waste time".

[chrome cypress]

Yes, I did.

[ember marlin]

...do you not see how those three statements together could constitute an insult directed at me?

[tired kettle]

If you're unwilling to accept limiting processes, unwilling to accept irrational numbers; then you simply cannot do any modern analysis, or any modern algebra. If you want to say "irrational numbers do not meaningfully exist, so I refuse to engage with them" you are wrong in the same sense that you would be wrong if you cut off your ears so that you couldn't listen to music.

[ember marlin]

I'm actually curious how much cutting off an earlobe actually would impair hearing.

After all, the inner ear would still be intact.

[gusty burrow]

"If by “in flux” you mean that pi is not constant, that is simply not true. Just because all of the decimal digits in pi are inconceivable to our human comprehension does that mean that they do not exist (look towards other fields of study such as quantum physics and general chemistry!)."

With all respect, "all the digits" of Pi is no different to talking about "all the natural numbers" in infinity. There is no conceivable fixed notion of such a thing, or quantity. In general chemistry, I see so place we use Pi that does not land it and make it rational, for uses in approximation. In the physical world irrationals do not exist, except perhaps in the paths of motion of bodies acting under a field of force if we assume a continuous distance metric in the force equations and of course a continuous time and distance metric in space itself. But such a thing is non-physical itself, it does not have properties in and of itself with being defined by the physical (the actual definitions of space or spacus in the root word is distance between, ie defined by the physical).

[tired kettle]

Then puncture the drum instead, and my analogy still holds.

[ember marlin]

I wasn't criticizing your analogy, just indulging a curious digression.

[gusty burrow]

I accept limits to infinity and that something can approach infinity in a function, same with irrationals used as limits in calculus. It is a convergent form of infinity to me.

[ember marlin]

"Convergent form of infinity" is literally incoherent.

[chrome cypress]

Not being willing to yield is a case of an impossible debate. I believe you fall into this category but I did not say that you were unintelligent.

I am not responsible for any other misinterpretations you have cultivated from my writings. But I will not speak any further, because, as I said before, there’s no point in arguing when both sides (not pointing any fingers) are unyielding.

[tired kettle]

So you accept that irrational numbers exist and can be the limits of a sequence of rational numbers? Then I do not see what the issue is here.

[ember marlin]

I'm willing to yield when I'm wrong, which does happen, but I'm not wrong when I say that pi is a number.

[terse palm]

great then just extend the line segment however long you want, THEN measure how long it goes from one corner of the square to another

[chrome cypress]

I agree with you.

pi is absolutely a real number.

[terse palm]

(btw it is possible to do it without overlapping but im supposing youre right for now)

[gusty burrow]

I still study them, I am not dunking on the usage of say real analysis and other fields in certain applications but those to me are always in the realm of approximation (which is how we navigate the real world with mathematical usage) as opposed to strict mathematics. And a lot of the time, some tweaks to certain definitions and we can have the same practical usage in my view.

[ember marlin]

https://youtube.com/watch?v=vV7ZuouUSfs

[tired kettle]

I assume they meant it's the limit of a convergent sequence of rationals?

[gusty burrow]

They are not numbers. Infinity is not a number either. We can use infinity as a limit, it does not make it a number.

[tired kettle]

I don't think anyone sensible would deny that approximations are a large part of real analysis. But what I dispute is your notion that, by involving approximations, it is somehow not "strict mathematics".

[gusty burrow]

Your responses to me have been replete with all kinds of fallacies and insults since the start of this debate.

[ember marlin]

Euclid requires the existence of irrational numbers.

It's such a simple construction, too.

[tired kettle]

So what? What's the significance of being a "number" versus a "non-number"? It's an arbitrary convention rather than a term with any significant mathematical weight, to consider this a number and that something else.

[ember marlin]

I mean, I don't think we can do arithmetic with non-numbers, can we?

[tired kettle]

You can do arithmetic with elements of finite fields, with points on elliptic curves, with rational functions, etc.

[gusty burrow]

That is not a line segment with a specific quantity or length. It is just an infinite line, ie conceptual or in flux. I do not know what translation you have read of postulate II, but you a greatly misinformed.

Considering your prior disrespect for Euclid and other great ancient thinkers, your lack of awareness of their work is no surprise.

[ember marlin]

No, shut up. The construction is so simple. Consider line segment AB of length 1. Construct point C such that BC is perpendicular to AB and of length 1. Then AC is of length sqrt(2).

[gusty burrow]

No, the word number has specific connotations going back to its root words in Latin and Greek. The axioms of Real numbers when it comes to operations are also only consistent with something that has specific quantity. If one argues it is impossible to operate on infinity and subract it from itself, then I find it inconsistent to say an irrational can be operated on likewise.

[tired kettle]

And what of your disrespect to Hippasus? Ignoring and belittling his life's work, and essentially justifying his murder?

[ember marlin]

Are you literally denying that pi is finite now? Is pi not strictly between 3 and 4???

[gusty burrow]

You do not even know Euclid properly, nor the difference between an axiomatic abstraction based on rules derived from rational proofs, and an actual proof measurement or quantity. You cannot talk me down, or change my mind when you miss the mark repeatedly on demonstrated clarity of knowledge on the works you cite, as well as comprehension of my reasoning on the matter.

[tired kettle]

Certainly historically and amongst laymen it has specific connotations. Within mathematics, as it exists today, I am personally unaware of any context wherein the distinction between a "number" and a "non-number" matters. Generally far more specific terminology is used nowadays.

[gusty burrow]

That does not make something finite.

[ember marlin]

YES IT LITERALLY EXACTLY DOES. X IS INFINITE IF AND ONLY IF FOR ALL Y, X >= Y.

[gusty burrow]

In most modern mathematics, yes. Not constructivist arguments and mathematical opinions. These things matter to me and others who care about clarity and mathematical principles.

[tired kettle]

I think you are misusing the word "finite". A real number is finite if it is smaller than some other real number. Perhaps what you mean is that pi "does not have a finite decimal expansion"; but this is also true of many rational numbers that you do not dispute, e.g. 2/7.

[ember marlin]

The person who says pi is not a number does not get to claim monopoly on "caring about clarity and mathematical principles".

[gusty burrow]

No, first of all you in your fit of rage on the keyboard are forgetting negative infinity. Second of all, that is divergent infinity. If we go to the root of the word infinity, it is perfectly acceptable to also have a convergent infinity assuming a non-discrete numberline.

[hidden jay]

If this chat continues there will be the first digits of pi displayed on the number of messages, this transcends all expectations

[ember marlin]

You're literally redefining every word in mathematics in your quest to deny a fact that every mathematician in the world has accepted for literally all of recorded history.

[gusty burrow]

You almost spilled my tea on my device. 😅

[ember marlin]

NOBODY GIVES A FUCK ABOUT YOUR FUCKING ETYMOLOGY.

[gusty burrow]

You really do get worked up over the possibility of someone having an opinion that differs to the consensus dont you? Perhaps you should look inward, and introduce some rationality into your methods of objections and discussion.

[ember marlin]

No, it's not you having an opinion I care about, it's you being wrong.

If you said you didn't like pi, I wouldn't really give a fuck.

But you said pi isn't a number.

And you are redefining both "pi" and "number" in order to try to make your case.

[gusty burrow]

It is not a number. It is a failure of measurement. 🙂

[ember marlin]

Say it all you want, that doesn't make it true.

[gusty burrow]

How are they not losing it?

[willow knot]

No I just thought it would be funny to snitch

By the way, Sapphire made squaring and rooting not commutative so √(2²) and (√2)² have different answers

[obtuse silo]

Crashing out over this is genuinely hilarious lmao.

[gusty burrow]

https://media.tenor.com/my38N82RLnoAAAAM/michael-scott.gif

[willow knot]

I do think its kinda lame to return to the friend group server to get validation

but it doesnt really matter

[gusty burrow]

https://media.tenor.com/JtXodwPxG0AAAAAM/meme-movie.gif

[tired kettle]

What I find funny about this entire conversation is that there are at least three professional, adult mathematicians who have wasted hours here, and you're probably in middle-school and really proud of yourself for getting us for this long.

Honestly, it's impressive commitment to the bit.

[gusty burrow]

No, I actually believe this. I just obviously find some of the responses amusing here enough to share.

[ember marlin]

You know the insane supervillain is the one saying that line, right?

[tired kettle]

Techie, it's not worth it to fight over this one.

[willow knot]

Hey, wasted is a little negative. I like to think that even fake sparring is worth something.

[hidden jay]

It’s a meme bruh

[ember marlin]

Who's fighting? I just think it's ironically appropriate.

Accidental self-awareness.

[tired kettle]

Not enough screenshots-within-screenshots. Here's another one.

[tired kettle]

Nice.

[ember marlin]

Wow, the quack has a sensitive ego, how surprising. /j

[willow knot]

Try not to sperg out over it

[gusty burrow]

No it is just a lame thing to do. Like wooo, I shared some funny moments from the debate.

Like you sharing it is just lame, like wow big deal people share stuff and joke about stuff.

Why bring it into the debate space?

[ember marlin]

Uh, could we not with the ableism?

[gusty burrow]

Probably the only thing we can agree on there.

[tired kettle]

That's not true. You both agree that it's really important whether irrational numbers are numbers.

[ember marlin]

Assuming we agree on the definition of the word "ableism", which seems unlikely considering you don't agree with common usage on the definition of any other word.

[willow knot]

I like to think anyone can sperg out

unless you're here to tell me you did it before it was cool

[gusty burrow]

True 😅

[gusty burrow]

It is a mathematics Discord with clear rules on certain language. It is ableistic. I did say when inviting you all to participate in the debate no offensive language (racism, ableism, transphobia or any ism or -phobia).

[willow knot]

I genuinely believe that sperging out isn't a disability thing, well okay maybe it is but in the socially disabled sense.

[obtuse silo]

What do you think the term “sperg” comes from?

[willow knot]

Aspergers? So what?

In some places, "spastic" means to act wacky.

[ember marlin]

...which is also ableist.

[willow knot]

And dont ask me about what the english call cigarettes

[willow knot]

Okay, you've interpreted this as ableist, now what?

Where is this going

[obtuse silo]

Do you think calling someone “gay” isn’g homophobic because people use it to mean [any arbitrary negative trait]?

Like, what’s the logic here lmao.

[willow knot]

I believe the word "gay" can be meant in ways that don't necessarily pertain to homosexuality.

[obtuse silo]

Yeah I’m done here 💔

[gusty burrow]

Just an absolute red herring to the debate. 🤦‍♀️

[willow knot]

Now I'm the one baiting

checkmate

[gusty burrow]

I hope they ban you for your antagonistic offensive language you cannot help using, you little rat.

[obtuse silo]

”little rat” is crazy

[crisp basalt]

ngl

the claim that no line segments and also no numbers exist would be way more self consistent than whatever rxrsapphire is saying

the concept that there is a line segment, and a corresponding number, that perfectly fits between (0,0) and (1,1) is equally contrived as the idea that for any line segment, there is a point on it, that splits it exactly, not approximately but exactly in half

[tired kettle]

-warn @willow knot Defending ableist language.

[surreal tide]

⚠ Warned markedoff

[willow knot]

soy

[obtuse silo]

It only exists when I mentally construct it smh.

[terse palm]

since when were they commutative...
sqrt((-2)²) = 2
(sqrt(-2))²=-2
sqrt(x²) is actually a definition of the absolute value function

btw fyi an operation not being commutative JUST means that the commutative property doesnt hold for ALL cases, it can hold for some, like how here it holds for all non-negative numbers (ie:the 2 you mentioned)

[gusty burrow]

Their knowledge in mathematics is really not that great. The only reason they are here is because I invited them into the discussion/server, in part to better understand my position but also in part to join the fun of debating something deep/intellectual as a challenge. They will not last long before being banned for using offensive language even though I told them not to, at this point they will.

[tawdry swift]

What in the living fuck is happening here?

And where the fuck is OP?

[willow knot]

Do square roots preserve signs then?

Could you maybe explain the second one a little?

[merry lava]

No

not necessarily

[willow knot]

All of this is also goes out the window because of imaginary numbers.

[lapis marten]

OP left the convo early on

But the devastation they've left in their wake is massive

[terse palm]

sqrt((-2)^2) is just 2 because (-2)^2 is 4, and sqrt4 is in fact 2 not +-2, because conventionally sqrt2 means the sqrt FUNCTION and for it to be a function it has to only give 1 output so we take the principal branch (the positive number)
ie:

sqrt(9) is 3 not +-3

sqrt(16) is 4 not +-4 etc

However sqrt(-2) is just isqrt(2)

so (isqrt2)^2 = i^2 * sqrt(2)^2 = -1 * 2 = -2

so thats why you cant always just cancel out the square and the root
bc like i said sqrt(x^2) behaves as an absolute value function so any negative x you input into it will become positive ie
sqrt((-3)^2) = 3
sqrt((-4)^2) = 4
etc

for (sqrtx)^2 you can just simplify it to x though (For real x) because its not the same as sqrt(x^2) since like I showed they aren't commutative

[devout stag]

what a way to waste time

[willow knot]

(sqrt(-2))² = -2?

[devout stag]

I recommend not using this thread

[willow knot]

Why, because it's cursed?

[devout stag]

sapphire is onto something very important and useful to mathematics

we should let them be

and not disturb them.

[terse palm]

Yes

[terse palm]

bc this

[willow knot]

I see.

Interesting.

So (√-2)² => (i√2)² => (i√2)(i√2) => (-1)(2) = -2

But √(-2²) => √4 => 2

[willow knot]

So is there a discrete order of operations for you to do them in?

I guess since the square root is enclosed it should be clear...

[terse palm]

Yes

[terse palm]

Yes but VERY important correction
-2^2 is -4
you have to specify with brackets (-2)^2
otherwise that sqrt(-2^2) would actually be 2i

[willow knot]

I guess I just meant commutative in the rationals? But I'm still wrong aren't I?

[terse palm]

Work from the inside out

[willow knot]

-2^2 = -4?

[terse palm]

Yeah this doesn't have anything to do with irrationals im pretty sure

Its more like theyre commutative over non-negatives

[willow knot]

I mean, not including imaginary numbers.

Fuck

I've made so many mistakes lmao

it's not even funny

what's the domain for non negatives?

[terse palm]

its ok dw
but you mean not including negatives?
bc you can do something like sqrt((-2)^2)
it doesnt include any imaginary numbers but still it fails that commutative property

[terse palm]

Wdym?

[willow knot]

Yes. That.

[terse palm]

yeah because -2^2 is typically read as -(2^2)
because order of operations PEMDAS
E for exponentiation comes before M for multiplication (by -1)
so you do 2^2 first then multiply by negative 1
rather than do (-2)^2

[willow knot]

Right. This is one of the reasons I don't like negative numbers.

Their notation is always inbred

[terse palm]

Thank god You meant don't like subjectively
I thought yall were denying negative numbers too for a moment...

[willow knot]

Well any negative number is essentially just 0-x

At least that's how it's sometimes parsed

Negative numbers to subtraction is like what fractions are to division

But with less organization

@terse palm Before I go, is there anything that IS commutative?

[terse palm]

I was afk sorry

[willow knot]

Nah it's fine

[terse palm]

Just like alot of irrational numbers are just logx or sqrtx but SOME people don't like that either...

[terse palm]

Addition and multiplication im pretty sure

[willow knot]

Honestly, it seems like most major mathematical disagreements emerge from an arbitrary lack of definition.

Computers are much kinder to me

[terse palm]

Hold up I don't want you to misunderstand by some im talking about a particular irrational denier, theres obviously like basically no disagreement about things like this these days im pretty sure

[terse palm]

But really you could come up with any function thats symmetric
(meaning f(x,y) = f(y,x)) thatd be a binary operation thats commutative
ie if i came up with f(x,y)=2^(xy)
for x = 1 y = 2 it gives 4, for x = 2 y = 1 it still also gives 4
so its commutative because no matter how you switch the inputs the output stays the same

[willow knot]

I do have curiosities but I should probably go to the discussion channel.

Should I ping you?

[terse palm]

sure ok

[supple spire]

Can we get this chat to 3141 messages

[hazy iron]

Also countable infinity not existing is wild lmfaoooo

@gusty burrow give me a bijection from ℕ to ℂ please

[obtuse silo]

That question presupposes N, which is countably infinite

They probably would disagree based on that immediately

[tawdry swift]

Is only N countable infinite?

Or is Z?

I know Q and R are not.

[obtuse silo]

Q is countable

Z is too

[tawdry swift]

Huh.

[obtuse silo]

Finite products of countable sets are countable

[tawdry swift]

Oh, comparison.

Yeah.

I need to up my set theory game.

[hazy iron]

Lmfao

@tawdry swift try proving that ℚ is coumtable

[sinful adder]

how do you see how many messages are there

[gusty burrow]

4 posts to a nice pair of Twin Primes. 🙏

[gusty burrow]

He was misusing the terminology, so in his context/argument it does not exist. Countable infinity has nothing to do with counting infinity.

[ember marlin]

22 isn't prime.

[gusty burrow]

And? There are more than 22 posts here. And I am actually offended you would think that I would make a mistake like that.

[ember marlin]

The first two digits of the number of posts is 22. When you said "a pair of twin primes", I thought you meant the number of posts would represent both primes simultaneously, i.e. the first two digits would be the first prime, the second two the second prime.

[gusty burrow]

LOL. Whu what? You never heard of ...? Nevermind.

[tired kettle]

This is reasonable in the sense that it's the same thing that I would assume. It's unreasonable in the sense that I am baffled that you are still engaging with this thread.

[obtuse silo]

hmm

@gusty burrow

What are your thoughts on the axiom of choice?

[gusty burrow]

How many of you are completely unaware of the Twin Prime Conjecture? 😅

Unbelievable.

[tired kettle]

What do you think the twin prime conjecture says?

[gusty burrow]

There can be no debate or subjectivity on the matter of what the conjecture states ...

[tired kettle]

There can easily be misinterpretation, which is why I ask.

[gusty burrow]

It is very clear.

[gusty burrow]

No. There cannot.

[tired kettle]

Then state it.

[gusty burrow]

Very bizzare and obvious request. The TP conjecture asserts the existence of infinitely many pairs of prime numbers (p, q) such that q = p + 2.

No subjectivity to that.

[tired kettle]

Bravo. However, under your mathematical framework, this is trivially false. After all, if infinity does not exist, there can not be ininitely many of anything.

[gusty burrow]

It is the simplest of conjecture statements.

[gusty burrow]

Infinity does exist. I never said it did not.

Strawman bs

[obtuse silo]

I mean, that doesn’t necessarily follow, no?

Even in finite set theory, you can prove there are infinitely many natural numbers, even though there is no set of them. There is also not in general any actual infinity(because there need not be any infinite set)

[tired kettle]

Jabs at you aside; this is indeed what the twin prime conjecture says. What is the relevance of this to your claim, on post 2262, that we are "four posts away from a pair of twin primes"? Techie read that as interpreting you as claiming that (22,66) was a set of twin primes; a reading that I frankly agree with. In what way does stating that 22 is not a prime indicate that Techie does not know what the twin prime conjecture is?

[tired kettle]

I cracked a joke and then my serious thing took long enough to type that my joke was interpreted as serious.

[obtuse silo]

ah, my bad then.

[tired kettle]

I am aware of how ultrafinitist philosophy works. I think it's silly, but I understand it.

[obtuse silo]

Just to be clear, ultrafinitism is “there is a largest finite number”, and finitism is “there is no actual infinity”, right?

If so, yeah ultrafinitism is quite silly.

[gusty burrow]

Why would someone with anything more than a simple understanding of mathematics state that 22 is a prime? It is an insulting assumption. I corrected him on multiple errors already.

[tired kettle]

Then why did you say this? #1350078306099269662 message

[gusty burrow]

He was misusing the term countable infinity in the wrong context as an argument. It has nothing to do with counting all of the numbers in infinity

[obtuse silo]

I assume they meant 2265 and 2267, no?

Wait the former isn’t prime

[gusty burrow]

No.

[obtuse silo]

Not sure then

[tired kettle]

2265 is divisible by 5.

[tired kettle]

...what?

What does that have to do with "4 posts to a nice pair of twin primes"?

[obtuse silo]

After they posted the image, the amount of messages was 2263

[tired kettle]

Ah.

[obtuse silo]

2263+4=2267 is prime and so is 2269

[tired kettle]

Yes, that makes sense.

I was confused about 2266 not being prime.

(Obviously; it's divisible by 2).

Okay, that clears that one up.

[gusty burrow]

It went to the wrong comment (the link).

[tired kettle]

No it didn't.

I just checked it again.

[obtuse silo]

I’m still a bit curious about this..

[gusty burrow]

Yes, it did for me. It scrolled up to the wrong comment.

[tired kettle]

Can you please click on the link from this comment, and reply to the message that it goes to? #1350078306099269662 message

[obtuse silo]

It goes here.

Well, that’s where what it replies to goes to

[gusty burrow]

It scrolled up to the comment below it. Can you stop arguing trivia? First my TP comment which was fine and now this. It did do that for me, not a fault of the link just a glitch in the scrolling.

[tired kettle]

Huh. This is weird. Maybe discord links are just broken.

[obtuse silo]

Sorry if that wasn’t clear, mb

[gusty burrow]

Sorry missed that one.

[tired kettle]

Your TP comment was bizarre. My followup question is; how does not immediately recognizing that 2267 and 2269 are prime indicate that we are "completely unaware of the Twin Prime Conjecture"?

[obtuse silo]

Oh lmao you said link.

Mb x 2

Thought you meant clicked what it replied to.

[tired kettle]

No. The message that I gave you a link to had a link within it. I wanted you to click on both of those links, and then reply to the message that you landed on.

[gusty burrow]

"The first post that would represent both primes simultaneously" ie he thinks it is referring to digit representations in pairs that are both primes, as opposed to their separation by 2.

[obtuse silo]

Yeah that’s what’s you tried to link, then.

Discord ain’t bugging, I an 🗣️

[tired kettle]

No, that is not what they think. They assumed that you thought we were four posts away from a number like 5961.

[gusty burrow]

Why would he assume that when someone literally right above posted the TT posts? He obviously does not know what the TP conjecture is.

He already made blunders on simple things about Euclid's work.

The Real Analysis stuff is forgivable, but that was not.

Especially him then assuming I do not know 22 is not prime 🤮

[tired kettle]

I interpreted your comment in literally the exact same way as I think Techie did.

You communicated poorly, as you are wont to do. Then you insulted someone for misunderstanding you.

[gusty burrow]

2267 is obviously likely prime even without knowledge of how to quickly determine that and 2269 not being a factor of 5 or 2 or 3 should allow one to quickly determine I was alluding to a p, q pair.

[white ingot]

What are y'all talking about?

[tired kettle]

How is that remotely obvious?

[tired kettle]

Nothing worthwhile. Begone from this thread, newcomer, before it sucks you in too.

[gusty burrow]

We are not 4 posts away from any number where the digits are ... 🤦‍♀️ why even argue this?

[white ingot]

How the hell did it devolve to primes

🤦‍♂️

[tired kettle]

Right. I read it as you being wrong about something weirdly easy. You've been wrong about plenty of things before, so it would be in-character for you; though it is a more obvious thing than you've been wrong about before.

[gusty burrow]

Jump to the top of the thread and see yourself.

[gusty burrow]

Like what? What have I been wrong on?

[tired kettle]

You claimed that pi is infinite.

[gusty burrow]

It is.

It is a convergent form of infinity.

[tired kettle]

No it is not. It is finite. It is, in fact, even less than 4.

[gusty burrow]

That does not make it finite.

is 1/infinity finite?

[tired kettle]

In that case, you are wrong about the definition of "finite". Or perhaps you have redefined most mathematical terms in a way that make you technically correct, but far outside of the common usage of the terms you are using.

[tired kettle]

Yes, for any sensible definition it would be zero. (e.g. arithmetic on the riemann sphere). Though, it is also often just undefined. It depends on the context.

[gusty burrow]

No, you are actually wrong. I have studied the roots of all these terminologies and the mathematical history of them. I am not just pulling definitions out of the aether or being whimsical. I speak Latin and Classical Greek and have read a lot of the works that led to our modern understandings and interpretations.

[tired kettle]

This does not make you correct. That's like if my brother said, "I'm not homophobic - the word actually refers to a bundle of sticks! I was calling them a bundle of sticks!". Contemporary usage is often informed by historical etymology, but it is not dictated by it.

Words can change meaning.

[gusty burrow]

No, it is not sensible to say it is 0 in an absolute sense. It is not a number, but it is not 0 either. If it is a function with a limit of 0, it will approach 0 but not reach it.  lim[x → ∞] 1/x = 0

[tired kettle]

I agree that $\lim_{x\to\infty}\frac{1}{x}=0$ does not imply that $\frac{1}{\infty}=0$. You'll note that that was not the sensible definition I cited; I instead cited arithmetic on the riemann sphere.

[vast spear]

Pear Category Theorem

[gusty burrow]

The modern usage of number has no clear meaning, at most we have the axiomatic requirements for real numbers which you cannot meet without invoking formalism and assuming quantity to something which incommensurable to it.

[obtuse silo]

There’s tons of things people call numbers in math that aren’t even real numbers either tbf.

Complex numbers, infinite ordinals, etc.

[gusty burrow]

The Riemann sphere is just more convoluted and involved way of getting lost on a chalkboard to prove something which is still in any tangible sense superfluous and irrational (in the non-mathematical sense of the word).

[tired kettle]

I agree that "number" has no clear meaning. Nor need it have; in modern mathematics it is redundant, and has been replaced by far more precise terms such as "rational number", "real number", "natural number", "ordinal", etc.

[tired kettle]

I would argue to the contrary; not only is the Riemann sphere neither convoluted, overly-involved, superfluous or irrational, it is in fact the natural setting for complex analysis, with a beautiful and elegant construction.

[gusty burrow]

Complex numbers, in their rational form, are still comprised of countable elemental units.

[tired kettle]

Though, I concede that we are now entering the realm of personal preference.

[tired kettle]

The complex numbers are uncountable.

[gusty burrow]

"Nor it need have"

[obtuse silo]

To be clear, sapph, do you mean “countable” in the linguistic sense of the word, or the mathematical sense(e.g. in bijection with a subset of N), or some other meaning?

[gusty burrow]

I said, in their rational form, are comprised of countable elemental units. That is my definition of a number. An individual number is comprised of or can be represented as countable elemental units in some form to produce measurement. Anything else is a failure of measurement.

[tired kettle]

Asking a contemporary mathematician for a rigourous definition of "number" is like asking for a contemporary physicist for a computation of the density of the aether, the medium through which light waves propagate. It is historically interesting, though ultimately not interesting to the contemporary subject itself.

[tired kettle]

https://tenor.com/view/breaking-bad-funny-wtf-wth-jesse-gif-17336046

[gusty burrow]

Definitions matter. The definition of a number matters. Formalism has destroyed common sense, operating on non-quantities that cannot be measured to begine with like they are numbers is nonsensical.

[tired kettle]

Claiming that definitions matter, and then arguing against formalism, is contradictory. The point of formalism was to attempt to rigourously define everything.

I agree that definitions matter, which is why it's concerning that your definitions rely entirely on other things that you have not defined, and refuse to.

[gusty burrow]

If you can represent something in some form as countable elemental units, you have measurement. ie you have quantity and can have a number. This is the ancient Greek and Latin definition of the world number |numeros| in its root word. Without that there is no specific quantity to operate on and meet the axiomatic requirements of a number.

[gusty burrow]

No it was an attempt to ignore and cast aside millenia of definitions to do make believe operations on letters on paper that have no relation to reality.

Constructivism is the only way for rigor to be maintained in mathematics, without it, it is no longer mathematics.

[tired kettle]

What do "countable elemental units" mean? It is a phrase that you keep repeating, and which I have never seen in a modern mathematical context aside from this conversation.

[gusty burrow]

It is the literal definition of the word number in Greek and Latin (the root of the word). ie we can represent 5 as # # # # #
We can represent a fraction 1/3 as # relative to # # #

It allows measurement. It gives something quantity.

Wihtout it there is no measurement.

Only in an axiomatic abstractive sense based on rules established from rationals beforehand there is "measurement" in such a sense of the word used today, that is not measurement.

If something has no countable elements to measure, there is no measurement.

It is no different to infinity, or more similar to it.

[tired kettle]

Are you utterly unaware that the definition of a word can change over time? The word "guy" is a rather well-known example of this; it evolved from just the name of Guy Fawkes, to refer to the effigees burned on Guy Fawkes' Day, to refer to people dressed in bizarre clothes, to eventually refer to men in general, to nowadays just being a casual gender-neutral way to refer to groups of people.

Just because the Greeks and Latins used a word in one way does not mean that we, two thousand years later, must do so as well.

[tired kettle]

And just to clarify, a number is a countable elemental unit, and a countable elemental unit is a number?

[gusty burrow]

There is no clear definition for number anymore. Mathematics must be based on fundamental truths, it is not something you can have wavy definitions that mean nothing tangible. The definitions and connotations matter.

Even in regards to mathematical definitions, the axioms cannot be met, because you cannot operate on a non quantity. You can subtract something that has no measurement or quantity from itself to make nothing. It is no different to saying infinity minus infinity. It is formalism at its worst.

[tired kettle]

I also take umbrage, actually, with this notion that ancient Greek mathematics was somehow less based on abstract imaginary bullshit than modern mathematics. You cannot draw a line with no width, despite that being how lines are defined in Euclid. Even the smallest point you can draw on a piece of paper must still have some area, miniscule though it may be. The geometry of Euclid is as imaginary as the famous cuts made by Dedekind; it is simply older, and more prestigious to history-obsessed highschool students who don't know any better.

[tired kettle]

Mathematics is not based on the definition of "number" anymore. It is based on fundamental truths, or at least on axioms and assumptions that we make that are at least as valid as Euclid's five.

[gusty burrow]

A countable elemental unit, would be in its whole form representation of things 1 or #. Without elements that you can reduce things to and count, you do not have anything to measure.

[tired kettle]

So the only numbers are the natural numbers, then? 1, 2, 3, etc.?

[gusty burrow]

Nope. Absolute false equivalence there. It is nothing like the Dedekind cuts and modern axiomatic abstractions applying rules established from rationals and measurable quantities to incommensurable things to apply operations on them.

A line, is not a line segment anyway. There is a difference. The former divergent and infinite, the latter has quantity.

[gusty burrow]

No, because we can represent rationals as whole countable elemental units. ie 1/3 can be represented as # relative to # # # we can still produce measurement.

There is an interesting neo-Platonic argument about this though.

[tired kettle]

A line, is not a line segment anyway. There is a difference. The former divergent and infinite, the latter has quantity.
This does not change my claim about how you can not draw a line, in the strict Euclidean sense. Euclid's lines are fiction, just as much as Dedekind's cuts are.

[tired kettle]

What of negative numbers?

[gusty burrow]

You cannot draw an infinite line in a physical sense, but you can assume one. There is nothing wrong with assuming an infinite divergent numberline. Just as there is nothing wrong with its reciprocal assumption.

[tired kettle]

Why is this assumption allowed, but not the assumption of quantities that can not be expressed as what you call "whole countable elemental units"?

[gusty burrow]

Divergent, ie from the origin point O, we can go in both directions on an infinite line. Infinity has no start point.

[gusty burrow]

Infinity is not a quantity.

[tired kettle]

I never claimed that it was.

[gusty burrow]

Well, the assumption is allowed, because even though it is not comprised of countable elemental units it is not claiming to be a number or specific quantity.

For example, if we take a limit or Pi, to me that is acceptable, just like a limit of infinity is acceptable.

[tired kettle]

Alright. Let's throw away rigour and formalism, and try to put this in terms that you will think are rigourous and formal despite not being so. Infinity is what happens when you go all the way down the number line for-ever and ever, correct? The square root of two, or pi, or whatever, refer to precise, specific points on the number line, do they not? In what sense are they infinite?

[gusty burrow]

Because they have no precise, specific points on the numberline. They are in flux or conceptual. They are constructed like a divergent function approaching infinity, but instead in the convergent sense.

There is no quantity or measurement.

Thus it cannot be operated on.

[tired kettle]

This is not their construction, not even in the Euclidean sense. Consider the following: Begin with a ray, whose origin is on the point A. One unit away on that ray is the point B. From point B, construct perpendicular to it, a unit segment on whose other end is the length C. Place the point of your compass on A, and the other end on C. From there, draw an arc, until it intersects your ray. It should intersect it at precisely one point D; if the distance from A to B is 1, then the distance from A to D will be sqrt(2).

[gusty burrow]

#1350078306099269662 message
#1350078306099269662 message

I already addressed this type of argument.

[tired kettle]

Those were each quite different arguments. I should know, because I made them. And I did not make this specific argument to you previously, because it didn't even occur to me.

Frankly, it was before I was able to understand precisely what you were arguing for. (You are not very good at communicating your ideas).

[gusty burrow]

#1350078306099269662 message << I address this type of argument here as well.

[tired kettle]

No you didn't.

"The existence of a number is tied to a way to construct it" "Here's a way to construct it, within the rules you've set forth (ancient Greek fetishism that puts Mussolini's Rome obsession to shame)" "I already addressed that, haha, I'm not addressing it"

[gusty burrow]

ngl the characterization was funny and did make me laugh a bit. But no, I am here for common sense, not to fetishize and adore Greco-Roman statues whilst larping in Greek robes and papyrus if that is your opinion of me.

Look to determine that AC = √2, and follows that AD = √2. You first have to assume that we are dealing in quantities and Pythagorean theorem can be applied in a way to yield one. It cannot, it is an axiomatic abstraction based on rules established from the rationals

Also the ficitious diagonal of a square whose side is of unit length, what we now denote as √2, was actually shown by Euclid to be incommensurable with its side.

[tired kettle]

I want to clarify. Your claim is that common sense holds that, if we draw a right-angled triangle two of whose sides have the same length; then the hypotenuse is a segment without length?

[gusty burrow]

There is no hypotenuse, at least one that does not overlap.

[tired kettle]

Do you dispute Euclid's first axiom?

[gusty burrow]

Not at all.

[tired kettle]

So, in the construction I have given, you must agree that we are able to construct the segment between the points A and C?

Does this not give a right-angled triangle ABC, with hypotenuse AC?

[gusty burrow]

No. Euclid’s postulates, including his first postulate were expressed without any reference to a coordinate system like our modern x‑ and y‑axes.

You are not thinking about it in the correct framework.

[tired kettle]

When did I invoke coordinates?

[gusty burrow]

By reification of the points as being part of a construct of a square, and then being continued in such an axiomatic sense for the line segment, you are invoking co-ordinate systems.

[tired kettle]

Given a ray whose origin is A, we can pick an arbitrary point B. On that point, we can construct a perpendicular. (This is a well-known construction). By drawing a circle centered at B and whose radius is the segment AB, we can intersect that perpendicular at the point C.

This is doable entirely within the Euclidean framework.

Unless you claim that we cannot draw a ray? Cannot pick a point on that ray? We cannot construct a perpendicular? We cannot draw a circle of given radius? That circle will not intersect the perpendicular? We cannot draw the line AC?

Where does this break down?

Which step, precisely, are we unable to do?

[gusty burrow]

There are no perfect circles, it is a ficitious construction that invokes infinity. Circles are irrational.

[tired kettle]

Ah. So it's Euclid's third that you dislike.

Good to know.

[gusty burrow]

In part, but it depends on the interperetations as well. The terms used with the connotations of the time are actually very interesting "locus" for example being one of them and its similar references in ancient Indian texts. In essence, at least in my view Euclid acknowledges the circle as infinite in his works.

[tired kettle]

Euclid's third axiom, according to Heath's translation according to wikipedia, states that we may, "describe a circle with any centre and distance".

I want to clarify. In all your appeals to ancient Greek mathematics; in all your arguing about how Euclidean axioms are the superior way to do mathematics, and that Hilbert and his formalism are bullshit; you also dispute the third axiom of Euclid's elements?

Not only is your worldview outdated; not only does your worldview cut you off from the beauties of contemporary mathematics; you even lack self-consistency.

[gusty burrow]

It depends, partially I do. But again, the inferrence of measurement is not necessarily there either. Euclid’s Third Postulate does not assume any form of numerical measurement. The focus is entirely on the ability to create the geometric locus of points at a fixed distance from the center. Hilbert took Euclid's work though and created an absolute monstronsity.

Where is the lack of self-consistency?

My argument is not "x philosopher/mathematician was 100% correct" anyway it is a strict constructivist argument, there is a difference.

[tired kettle]

At this point I am done. You have advocated for self-defeating, self-contradictory, ill-defined points. You have insulted and belittled people for not recognising your genius.

If you want to completely cut yourself off from modern mathematics in order to advocate for your Euclid fetishism but not axiom 3, then be my guest. I'm not wasting any more time on this.

[gusty burrow]

I have barely mentioned Euclid in elaborating on my position, only when correcting other people's incorrect arguments misusing Euclid's work as arguments (not referring to you specifically as many others before reference Euclid incorrectly as an argument). My argument is rooted moreso in Pythagorean/neo-Platonic arguments put forth by other figures as well as modern constructivists who refuted formalistic notions.

I have not belittled anyone who did not use the same language against me multiple times beforehand.

[devout stag]

https://tenor.com/view/sound-the-car-alarm-cat-gif-9895924581726852336

so when are we closing this thread

[gusty burrow]

When it reaches 3142 posts

It has a wealth of resources and arguments for anyone interested in the arguments for/against on both sides.

I have offered my best and so have other people.

[devout stag]

it has nothing useful lol

[gusty burrow]

It deserves a shrine. And a marble statue.

[devout stag]

it contains other people's tries (in vain) to convince you

[gusty burrow]

It has the best arguments for/against the entire philosophical enquiry. And also the worst.

[devout stag]

that's about it

[gusty burrow]

Those are counterarguments. And I responded to all of them.

People can look at the debate and come to a conclusion.

It has the best arguments for and against.

Constructivism vs Classical/Formalism

And it is not at 3142 posts. Which would be fitting.

[devout stag]

I don't see any use for any of this personally

but I'm on the far end of the hating-philosophy spectrum

Don't worry we won't actually be closing this thread forcefully that's against the rules probably

[gusty burrow]

I have received multiple DMs just because of this debate for different reasons, people are interested in the discussion.

It is thought provoking to many.

[devout stag]

who dmed you?

Interesting why they won't discuss here.

[gusty burrow]

I do not share personal DMs to me without permission. A lot of the time, they do not probably want to be seen talking to "the crank" in the village hall so to speak I would assume. Mostly it is people who want to know pose an argument not addressed here, or curiosity of how I would intepret something or implications on calculus (like people thinking I would want to do away with limits because of it etc.).

[devout stag]

👍

[gusty burrow]

There have been negative reactions to people engaging with me in a civil way by others so it is no surprise some prefer to keep it private. Which is fine with me.

[devout stag]

have you ever tried this in a philosophy server?

[gusty burrow]

Yeah not many mathematicians on them or people interested to my surprise, you do not get much engagement at all if any. And nobody who disagrees will give you strong arguments either. I want the best arguments people can come up with against my position.

[devout stag]

maybe they are ignoring you because they don't think it's worthwhile

didn't you turn down all the arguments with nearly undefined terminology?

[gusty burrow]

No, not my impression. They are just not mathematicians. People who study philosophy and lurk in those Discords tend not to be studying mathematics also at any high level or deeper level/interest.

[gusty burrow]

The entire argument is around terminonology. What is a number? And what can meet the axiomatic requirements?

[devout stag]

but your arguments are philosophical

not mathematical

you are debating over axioms

[gusty burrow]

They are both. Have you heard of constructivism? It is a part of mathematics and mathematical discourse.

[devout stag]

have you worked on any mathematics? apart from whatever this is

[gusty burrow]

Yes of course.

[devout stag]

constructivism is a philosophical ideology I'm preety sure

which stems from math maybe

[devout stag]

wanna share?

unlike this, there won't be any ground for debate there

[gusty burrow]

No it is a part of mathematics. Modern mathematics is rooted in philosophical discourse around numbers and notions of formalism, classicalism etc.

[devout stag]

very sure it isn't

[devout stag]

so wanna share?

[gusty burrow]

I am not sharing my mathematical work on here. Someone could take my work and there are prizes for the things I am working on.

If you are talking about discoveries.

As opposed to just, well mathematical exercises.

[devout stag]

you just went up the crank index

[gusty burrow]

Cool, thank you for your intellectual engagement.

[devout stag]

I'm just extremely interested in how your philosophical stance blends in with your mathematical work

gatekeeping groundbreaking discovery is wild

@gusty burrow try uploading it on Vixra.

[gusty burrow]

They are not complete yet. And gatekeeping is not the same thing either.

[gusty burrow]

My mathematical work is not in somewhere that requires assuming irrationals are fixed quantities anyway. It is in analytic number theory mostly.

And physics related work that has mathematical proofs.

[devout stag]

what's wrong with sharing?

you always have proof that you came up with it first

[gusty burrow]

I disagree, if someone takes the final condition that needs proving to prove a statement, when you discovered the condition itself, they still get recognition as the person who proved the statement.

[gusty burrow]

But I can DM you some other work and discoveries if you like 🙂

That are math related, but also physics related.

[devout stag]

yes, please do

[gusty burrow]

okie 🙂

[merry lava]

I'm guessing it works sort of like modular arithmetic ?

[terse palm]

Btw rx so do you think you can't measure the length from one corner of the square to the other?

[tawdry swift]

Correspondence with the naturals, right?

[chrome cypress]

That sounds quite reminiscent regarding the proof of a certain, French conjecture! I hope we will have the opportunity to see some of your work once you have formally submitted.

[hazy iron]

Yeah prove that you can draw a bijection between them

[gusty burrow]

No you cannot. There is no measurement of such a thing. To measure you must have specific quantity and something comprised of countable elemental units in some form or metric.

[ember marlin]

Euclid says you're wrong.

[crisp basalt]

as everyone's favourite manager i am advising everyone to stop discussing this, certain people with certain beliefs will never change those beliefs

[gusty burrow]

You have already misquoted/misinterpreted Euclid multiple times now.

[obtuse silo]

We should just make a thread inside the thread

This discussion isn’t actually on topic

[gusty burrow]

"irrationality of Pi"

"it's just something fundamental."

"*We know that an irrational number cannot be expressed as p/q (p,q are coprime integers). Then what is to say about π? After all it is the ratio of the circumference and the diameter of a circle. *"

[terse palm]

So what happens when you place a ruler on it and measure it?
Also can you please define the whole countable elemental units thing and justify why it's a requirement for something to be measured/ a number? Since like the flux argument you seem to keep using it without defining/justifying it

[merry lava]

yes, if the diameter is rational, the circumference is not, so the fraction isn't of two rational numbers, if the diameter is irrational, then the fraction isn't two rational numbers, so either way, it's irrational

[obtuse silo]

I mean, this discussion isn’t relevant to the thread. Pi isn’t a rational number, which I assume we can agree on, right?

Dicussion about the existence of pi isn’t relevant to whether it’s rational or not(which is what they were asking)

[rough rapids]

the discussion hasnt been relevant for the last thousands of messages

is there really much point bringing that up now

[obtuse silo]

True.

[rough rapids]

and it seems to have died down now anyways

[gusty burrow]

*Also can you please define the whole countable elemental units thing and justify why it's a requirement for something to be measured/ a number? Since like the flux argument you seem to keep using it without defining/justifying it * << I have explained this multiple times now, even to you in replies.

#1350078306099269662 message

[gusty burrow]

Pi is not a ratio of a diameter to the circumference, because circles are fictitious objects and have no ratio to anything at all. They are irrational themselves unless one has been hoodwinked by the worst notions of formalism. Pi is not a rational number, it is not an irrational number either, it is not a number at all and likewise if something is not rational it is not a number.

[gusty burrow]

If you believe that 2rPi = Circumference, outside of approximation purposes you are inferring ratio in some formalistic sense if you think it is alright to perform such operations on non quantities. That was the OP point. Their layer points extended into the discussion about Pi having specific quantity.

[obtuse silo]

You know, this reminds me of a quote by Kronecker

”What good your beautiful proof on π? Why investigate such problems, given that irrational numbers do not even exist?"

[gusty burrow]

It is exploring a convergent form of infinity or infinite function. It can offer, when encoded in different formats, ways of producing reproducable infinite streams of non-uniform data.

Even the rational results when we land them are interesting with intrigueing properties depending on changes to n.

[gusty burrow]

Just entered the thread/debate?

If anyone by the way is just reading this and does not want to read through the entire debate to understand the constructivist/neo-Pythaogrean/Platonic position I advocate on behalf of.

I am happy to DM you to a full complete summary of my arguments all together.

[supple spire]

Just send it here (My goal is to inflate the message count)

[gusty burrow]

No it could be seen as spam as it is a wall of text.

[supple spire]

PDFs I suppose

[terse palm]

your definition for countable whole unit elements is literally just multiples of rationals, so youre saying rationals exist bc theyre multiples of rationals

and the flux thing literally disqualifies 1/3

[gusty burrow]

1/3 in base 3 = 0.1 you cannot do that in any base system for an irrational though without reification, begging the question and serious formalistic fallacies.

And rationals exist because you can express them relative to others and the single elemental unit. ie 1/3 is # relative to # # #. We can count and produce measurement.

[ember marlin]

Pi is exactly 10 in base pi.

[gusty burrow]

No such thing as base pi my friend. 🤦‍♀️ 😅

"... without reification, begging the question and serious formalistic fallacies."

[tired kettle]

I don't see why not. Surely given a real number $x\in(0,1)$ we can find a sequence of integers $0\leq a_{n}\leq3$ such that $x=\sum_{n=1}^{\infty}\frac{a_{n}}{\pi^{n}}$. You probably just modify the proof that base $p$ works for integers $p$.

[vast spear]

Pear Category Theorem

[crisp basalt]

^

[gusty burrow]

"... without reification, begging the question and serious formalistic fallacies."

[tired kettle]

My bad. Should I just lock the post?

Fuck it, why not?

[tired kettle]

The answer: Because then rxr starts flooding #geonosidan-prison-complex.

[gusty burrow]

https://c.tenor.com/RKJ_WYyQd-UAAAAC/wee-woohoo.gif

yay.

[gusty burrow]

In the usual setting for base-b representations, b is an integer greater than 1. The classical algorithm (often called the “greedy algorithm”) and the uniquenes proofs actually rely on arithmetic properties of integers.

The “Exactness” and Constructibility Issue

• When someone claims “Pi is exactly 10 in base pi,” they are attempting to reify the number π by using it as both the base and its own representation. In analogy, in any numeral system the symbol “10” represents the base itself. In an integer-base system, this works because the base is an accepted, fixed, and finitely describable whole number.

• However, using π a number already only defined as a limit of an infinite process as the base raises concerns. The representation “10” in base π would mean that the number π equals 1·π¹ + 0, which is trivial by definition. But the system as a whole would require all numbers to be represented by infinite serie using π as the denominator. For constructivists, this does not resolve the “landing problem” aforementioned because you are still encoding numbers (even apparently “simple” ones) by an infinite, indirect process.

By trying to generalize the representation theory from integer bases to the irrational base π (or any irrational β), you are “reifying” the infinite limit process. That is, one treats the limit (the number represented by an infinite series) as if it were a finished, finite object - even though by the constructivist criterion it never has landed fully. Simply extending the familiar algorithm to an irrational base does not resolve the underlying philosophical objection brought forth.

[prime glade]

It is just advice! Maybe you can learn something yourself from the discussion, even if she’ll not budge.

[hazy iron]

Yes lock this thread lmfaooo

I love how rxr isn’t even the OP lol

[gusty burrow]

The OP agrees with me. 🙂

@hazy iron You have no argument to refute strict constructivism.

Thus you would rather shut it down.

[ember marlin]

This is not remotely "strict constructivism". A mathematical constructivist rejects non-constructive existence proofs. Meaning you actually should reject the existence of infinitely many primes, but at least the computable numbers are fine.

[gusty burrow]

You clearly have not read up on what constructivism is properly. Euclid's proof of infinitely many primes is not rejected by a constructivist approach, to claim so either shows a.) a misunderstanding of constructivism or b.) an attempt to strawman

[tawdry swift]

💀

Can we please stop fighting?

[ember marlin]

The proof does not construct the set of primes.

If constructivists accept it, that's only them being hypocrites.

[gusty burrow]

Constructivism insists on proofs that not only show an object exists but also provide a method or algorithm for constructing such an object in its claimed form.

In Euclid’s proof, given any finite list of primes, the construction of a new number (the product of those primes, plus one) and then showing that at least one prime divisor of that new number is not in the original list follows a clear algorithmic recipe. This procedure explicitly produces an additional prime, aligning with constructive principles.

You are misunderstanding constructivism as some sort of complete rejection of all mathematical proofs that are not constructed in a specific example for all cases, to imply a false dichotomy of philosophical choice of either being opposed to any and all mathematical proofs or formalism/classicalism being the only way to accept them. This is a misrepresentation of what constructivism means.

If that was true, Pythagorean theorem would not apply beyond a few select triangles.

Non-constructive proofs rely on indirect arguments such as proof by contradiction combined with existence axioms, Euclid’s approach is constructive because it doesn’t merely assert “infinitely many primes exist” via contradiction and axiomatic abstraction; it gives a finite algorithm to extend any given finite set.

The objection that constructivists sometimes reject non-constructive existence proofs is not meant to dismiss all proofs concerning infinity. Instead, it targets those that fail to offer a method to "construct" a particular instance. Euclid’s proof, by contrast, does exactly that.

[ember marlin]

Okay, so by exactly the same logic, we have algorithms to produce arbitrarily many digits of pi, so pi must exist and be a number.

[gusty burrow]

By producing digits and expanding into the decimal system your answer, you are landing a rational number and creating a measurment, which is a failure of measurment of Pi itself. You have landed it already so to speak with the outcome of the sequence with a given n value.

[supple spire]

Anarchy

[ember marlin]

How can to "fail to measure" a thing that doesn't exist?

[tired kettle]

This would only make Euclid's proof constructive if $2\cdot3\cdot5\cdots p_{n}+1$ were always prime. It is not. Consider $$2\cdot3\cdot5\cdot7\cdot11\cdot13+1=30031=59\cdot509.$$

[vast spear]

Pear Category Theorem

[tired kettle]

It is a proof by contradiction, demonstrating that there is no finite list containing all prime numbers by arriving at a contradiction - if there were such a list, there would be a prime number not on that list. (Either that product plus one, or some divisor of that number). It does not provide a method or algorithm for generating infinitely many prime numbers; nor even for always finding a new prime number given a finite list of prime numbers.

[gusty burrow]

Glad you are finally seeing that irrationals are not numbers or quantities.

[gusty burrow]

I know how Euclid's proof works. I did not write out the entire proof.

[ember marlin]

That is not what's happening, that's me pointing out an inconsistency in your position.

[tired kettle]

Mayhaps the counterargument is - well, this isn't how we get Euclid's list. The algorithm clearly lays out that $p_{1}=2$, \$p_{n+1}=1+\prod_{k=1}^{n}p_{k}$. In this case, $p_{1}=2$, $p_{2}=3$, $p_{3}=7$, $p_{4}=43$ and $p_{5}=1807=13\cdot139$ is once again not a prime number.

[vast spear]

Pear Category Theorem

[tired kettle]

Right. I'm not debating you on the merits of the proof, or whether it works. I'm refuting your claim as to the nature of the proof. It is not an algorithmic, constructivist proof.

[gusty burrow]

Algorithmic Recipe to Generate a New Prime:

• Start with any finite list of primes, say [p1, p2, …, pn].

• Compute the product P = p1 × p2 × … × pn.

• Form the number N = P + 1.

• Even if N itself is composite (as in your example with N = 30031), every prime divisor q of N satisfies q ∤ P. (None of the pᵢ divide N because each pᵢ divides P, thus pᵢ does not divide P + 1.)

• Therefore, at least one prime factor q of N does not occur in the original list.

This process provides an explicit method to generate a prime that is new relative to any given finite list. Even though the number P + 1 may not be prime, the proof gives a concrete, finite procedure:

Given any list, you can compute P + 1 and then use a reliabble factorization algorithm This confirms, step by step, the existence of a new prime and thus an algorithm to extend any finite list of primes.

About the contradiction aspect, if we notice that the structure of the argument demonstrates that for any finite set L of primes, one can construct a candidate N and subsquently find a new prime factor q ∉ L.

This is not merely a pure non-constructive existence proof. It gives you a method by which, from L, you explictly get a new prime. Thus, it still walks the constructivist line because at no point does it assume the existence of an infinite completed set; it only shows that any finite approximation can always be extended. There is a clear difference.

The initial list of primes, is calculated beforehand algorithmically from finite objects, not assumed.

Infact the Euclid proof is actually referenced as a good example of a constructivist proof.

[crisp basalt]

the problem is that this proof is almost always presented as a proof by contradiction

which of course it doesn’t need to be

[ember marlin]

Doesn't it?

[crisp basalt]

though can you constructively prove the fundamental theorem of arithmetic?

it’s not necessary here because you can prove there is some prime factorization

[tired kettle]

I suppose my followup question is; given an arbitrary composite number, how do you actually find the prime factors? This is a very difficult problem that underlies much of modern cryptography. I might be mistaken, but saying "yep, the prime factors exist" without being able to actually point to them doesn't strike me as being particularly in-line with your constructivist philosophy?

[gusty burrow]

Every composite number, by definition has prime factors. There is not reification either.
You make composites from primes and their products.

And lol as for mathematics in general anyway, obviously have no problem engaging in different things whatever they might be. If someone gives me an algebra test, I am not going to tear it up out of principle and storm out the test because it has a square root that is incommensurable which needs operating on. 😅 I just believe in distinction between approximation mathematics (for example continuous curves, integrals etc) and strict mathematics.

[obtuse silo]

https://math.stackexchange.com/questions/30127/is-there-an-intuitionist-i-e-constructive-proof-of-the-infinitude-of-primes

[obtuse silo]

I think “there’s finitely many for any given n so you can just check them hypothetically and see there is a prime” usually works fine(at least, for Brouwer iirc?)

[tired kettle]

Yeah but that's O(n!) and thus kind of gross. I don't like it.

[crisp basalt]

actually, can you? one step in the argument i looked up involves saying "either n is a prime or not", which shouldn't be legal

[obtuse silo]

Pretty sure that’s fine for natural numbers(because you could in principle check every number below p algorithmically or whatever and see if it is prime, or not)

[crisp basalt]

it is not constructively valid to say that every subset of a singleton is either empty or a singleton

it could be that you can do this with N because "you can theoretically check all N but you can't check all singletons" (??)

[merry lava]

This makes me wonder if you had a function, the smallest prime factor of the prime factorial(n) +1

I wonder what it's graph would look like

[white ingot]

where pi?

[merry lava]

Oh yeah, the smallest prime factor of a number is just the smallest factor besides 1

[white ingot]

you think the formula for primes is really simple or really complicated?

[obtuse silo]

Yeah

There’s a proper class of singletons

🥶

[crisp basalt]

nobody has checked all of N though

i don't see why it's different

i don't think it's constructively valid to say that every subset of {{}} is {{}} or {}

[obtuse silo]

Idk

[crisp basalt]

such a proof is gonna end up having some casework on the truth of a statement and that's not valid

[obtuse silo]

MSE post seems to agree(about checking all the finitely many numbers before p), fwiw 🤷

[crisp basalt]

idk if i believe them

[merry lava]

Well, there isn't just one formula
but I think if there was a simple good fast formula it would be well known by now

[white ingot]

There could be the thought that we haven't developed the tools for that yet

[prime glade]

Are... are you being serious, Techie?

[prime glade]

yes.

I don't know how constructivism is so misunderstood here. You can obviously construct, for any finite set of primes indexed $p_1,p_2,\dots, p_k$, a prime not in that set $P$ by just calculating the prime factorization of $p_1\cdots p_k + 1$--this is in response to techie

obviously there are intuitionistically valid proofs of the fundamental theorem of arithmetic because they don't necessarily rely on the PEM or any other constructively dubious principles like Markov's principle.

[vast spear]

magma the chaos magmician

[gusty burrow]

From a constructivist point of view, FTA is not merely postulated by an axiom of completeness but is proven by an algorithm that finite steps yield the prime factors (as I have put before “landing” them on the number line, so to speak) and by an elementary, constructive argument ensuring uniqueness.

This aligns with the insistnce that to be counted as a “number” (or a property of a number), one must obtain a finitely realizable, discrete construction rather than only an indefinitely convergent process. I am actually writing out a complete FTA into 4 steps.

[gusty burrow]

For some reason people do not get it, they think it means or implies an absence of mathematical proofs. If that was the case and it was that stupid/poorly thought out, constructivism would have no proponents at all as it would preclude all mathematics. 😅

[prime glade]

Well, lots of things are constructively invalid.

[gusty burrow]

True, but if the fundamental theorem of arithmetic or proof of infinite primes was precluded by taking a constructivist stance, nobody would have ever adopted it in the first place. That said, my perspective/ideology differs from most of traditional constructivism in some aspects anyway. I feel like in some areas it does not give enough leeway for proofs and in others it is just accepts and bends the knee for formalism/classicalism when it should not.

[prime glade]

you can prove the existence of a prime factorization by induction

this may somehow shock you guys but induction is constructively valid!!!!

[obtuse silo]

It shouldn’t be ✍️

Very fishy !

[crisp basalt]

does this
respond to the thing i raised

[prime glade]

but uh, not all instances of "P v ~P" aren't intuitionistically not affirmed

i guess im curious, do you think $\forall x (x = 0 \lor \neg (x = 0))$ is constructively provable over the integers

[vast spear]

magma the chaos magmician

[crisp basalt]

i wouldn't guess so if you weren't asking in this specific scenario

[prime glade]

yeah so not all predicates are undecidable

[crisp basalt]

saying some integer is either 0 or not zero seems just a bit harder to show than saying that some subset of {{}} is either {} or {{}}

[prime glade]

this is kind of important to make clear

well intuitionistic set theory and something like heyting arithmetic are pretty different

but yeah so the fta is intuitionistically provable i want to make clear

just to make absolutely sure i wasn't lost in the sauce that "X is prime" is decidable since it is Sigma_0

uh, im honestly a bit perplexed as to where this confusion comes about

okay guys so just to be clear, it is also inconsistent with minimal logic to affirm ~(P v ~P)

this is different when you introduce quantifiers

well, unbounded quantifiers at least

since now de morgan fails intuitionistically

i guess to summarize so you guys aren't confused by my incompetent ability to communicate

uhhh

yes, the fta is intuitionistically provable

[crisp basalt]

x is prime could be in the form of some negation like for all a ≥ 2, for all b ≥ 2, ab ≠ x, and then "x is not prime" would be like "it is not contradictory that there exists a ≥ 2, b ≥ 2 with ab = x" and then we can conclude that either x is prime or not prime, but then afaict we can't conclude that actually they do exist which we need to do later when establishing the factorization

[tired kettle]

I support constructivism with the added constraint that things have to be polynomial time in particular. O(n!) algorithms or whatever are fucking stupid. I have seen no valid proof that Groebner bases exist.

[prime glade]

so like, when you're formalizing "x is not prime" do you want to write it as like "~(phi -> \bot)" for some phi or what

[tired kettle]

This is the bad-math-phil shitposting channel at this point, so I decided to lean into it.

[prime glade]

anyhow all of these sentences are Sigma_0 so they are decidable in HA

[crisp basalt]

what is that

[prime glade]

google the "arithmetical hierarchy"

[tired kettle]

Ping @molten lichen

[prime glade]

the fta is at most Pi_2 so it is also definitely provable in HA since it is provable in PA

specifically, PA is Pi_2-conservative over HA

[crisp basalt]

automagical negation of sentence rephrasal detected

[prime glade]

i asked this, you never responded

[obtuse silo]

Pi_2(S^2, S^1) ✍️

[crisp basalt]

idk what bot is

oh bottom

[prime glade]

$\bot$

[vast spear]

magma the chaos magmician

[crisp basalt]

well so the thing i've done here is "x is prime" looks like phi => bot, and so "x is not prime" would of course be (phi => bot) => bot

[prime glade]

alright, cool

[crisp basalt]

which is true iff phi => bot is not true, iiuc

[prime glade]

yeah so it doesn't actually matter to be clear

since (phi => bot) => bot is still going to be deicdable

but yeah you will rewrite it to something which is easily identified in the arithmetical hierarchy

[prime glade]

anyhow this is the most general answer i can probably give?

this just means that every Pi_2 formula provable in PA is provable in HA

due to friedman

well, at least that's uh who it's named after

[obtuse silo]

Intuitionistically, $\forall{P}(\neg\neg{P}\vee{\neg{P}})$ still holds right?

[crisp basalt]

this is heyting algebra?

[prime glade]

idk actually who first proved it

[prime glade]

heyting arithmetic

[vast spear]

Topology & Groupoids

[prime glade]

just the intuitionistic version of pa

it's literally just PA + IQC

that's it

idk man i guess you can insist HA is not at all constructivist

that's fine

kind of weird but whatevs

[prime glade]

uh, no

i spent some time trying to construct a countermodel

~~P v ~P is the wlem though

IPC + wlem is known to be consistent and subclassical

i found some cool program that doesn't work though

called heytinget

[prime glade]

so guys what did we learn? constructivism is not what you guys thought it was

guys if the fta was not constructively provable, wtf do you think is constructively provable

[prime glade]

yes, indeed you can constructively prove sin(x) has a smallest positive root

you can do a large amount of analysis constructively though you do need to change the definitions a bit

in ways which aren't just "godel-gentzen translation"

[prime glade]

alas, constructivists will still get no more respect after this

it's just philosophical mumbo jumbo or whatever

[crisp basalt]

.-.

can we be not condescending

[prime glade]

alright, sorry

it's just 99% of discord people think constructivism is absolutely ridiculous and then know nothing about it

i am genuinely sorry

[crisp basalt]

i thought this was true because !P => !!!P, and !!!P => (P => !!P => F) = !P, but

maybe not

yeah i should have realized constructivists weren't just fooling around with statements and stuff and probably had an axiomatization of N also

[ember marlin]

I think Sapphire is absolutely ridiculous and not an actual constructionist, which is why I was arguing against their position.

[prime glade]

if you are curious about constructive analysis, bishop is the guy who is attributed with really getting the whole project started though lots of work was done by people like brouwer before him, there's his foundations of constructive analysis then there's a revision of it by douglas bridges called Constructive Analysis which is pretty good

im also a bit biased because douglas bridges has Representations of Preference Orderings which has a great chapter on jointy continuous utility functions

[prime glade]

alright, i misunderstood you then, my apologies

but i wasn't really so much complaining about you, this is a general trend ive seen among math people

[crisp basalt]

i've accepted that sapphire's position is probablyyyy self-consistent, and #1350078306099269662 message reasonable

[tired kettle]

Constructivism is absolutely ridiculous. If it weren't, I'd know more about it.

[prime glade]

haha

i mean you can always learn a little bit of logic, i promise it is not hard otherwise i wouldn't know any of it

but there is some degree of carefulness i think necessary that makes it hard to give a simple rundown of

[tired kettle]

All the logic I know, I learned from watching Ben Shapiro compilations on YouTube.

[prime glade]

haha

[prime glade]

I think the biggest problem has just been that she tends to use a lot of terminology in very, very nonstandard ways.

Now I don't know if I agree so much the position is reasonable.

I'm a bit undecided on whether it is self-consistent, too, but I'm willing to grant it is at the cost of it being very unmotivated!

But definitely, the natural numbers are much more philosophically secure than kind of arbitrary real numbers.

Hell, even constructible numbers can be screwy.

[prime glade]

And when Q can be defined in such a simple way from N in terms of very simple operations on pairs (p,q) of integers, it's easy to see the allure of not going any further.

However, I think she does go into crank territory when she begins to talk about physics.

I only recall a bit, though.

I guess one problem I have is that I don't really see any robust reason to privilege the integers ontologically over say, compact groups.

It's not like fictionalism is unheard of, though its success is... well, who knows!? (I'm looking at you, Field)

[tawdry swift]

Augh

[ember marlin]

"Irrationals are not numbers because bla bla 'failure of measurement' bla" is "self-consistent"?

[prime glade]

Yeah, so how is it inconsistent?

Like, maybe absurd or whatever. But where's the internal tension?

[ember marlin]

Frankly it's not defined well enough to evaluate its consistency.

[prime glade]

Okay, so for sapphire we have these idealized objects which we can specify in a variety of ways "perfectly round," for instance, "spherical," et cetera, which do not exist, but we can talk about them being able to be approximated in a variety of ways and irrational numbers are kind of abstractions of contextual levels of approximation by existent quantities which are rational numbers.

Now, as you pointed out earlier, one glaring problem is we still want to say whatever whatever we're approximating exists. Otherwise, how else could we approximate it?

Now this is kind of a more general problem though.

Like, let's say I want to say "you look like Sherlock Holmes," does this commit me to the existence of Sherlock Holmes?

[obtuse silo]

Cause God gave us the integers per Kronecker ✍️

[prime glade]

Haha, yes she did.

She actually only gave us N, smh

[crisp basalt]

god didn't give us the integers else the integers wouldn't be finite first order undefinable

[prime glade]

thank you, thank you bro

[obtuse silo]

Fun fact! Every statement about rationals is actually a statement about integers. Same with reals to integers(a real number is just a sequence of integers)

Only the integers exist

[prime glade]

So true.

[obtuse silo]

Real numbers are just shorthand

[prime glade]

Aren't you topology-pilled?

[obtuse silo]

The integers in the Furstenberg topology are homeomorphic to Q too anyway

[prime glade]

💀

Yeah, I forgot about that, it's pretty neat.

[obtuse silo]

Iirc it’s a topological group on Z, too

[prime glade]

well, wikipedia says it's the topology induced by the inclusion in profinite Z

with profinite topology

[tired kettle]

At this point, the question is not whether we are committed to the existence of Sherlock Holmes, but what existence even means. Certainly we can say that Sherlock Holmes exists as a popular character in our collective consciousness, but not as a real person.

[prime glade]

so i can buy that

[prime glade]

Metaontology moment!?

Alright, sounds fire to me.

I'm down.

Let me get something to drink real quick.

[tired kettle]

Okay so I'm starting metaontological but I'm going to quickly veer into a sort of strange status-quo preserving pragmatism where I end with "existence doesn't matter, but how we use these concepts does regardless of whether they exist or not".

[prime glade]

Alright, I see.

[obtuse silo]

Nothing exists

[prime glade]

Okay, well what do you mean by that?

[obtuse silo]

Existence is a scam pushed by big $\exists$

[vast spear]

Topology & Groupoids

[prime glade]

Do you mean it in the Westerhoffian sense or the Priestian sense?

I am, of course, jesting.

[prime glade]

Yeah, I mean I think this is pretty intuitive, actually.

[tired kettle]

This feels vacuously true.

[prime glade]

So, yeah, before we get into metaontology I think it's worthwhile to ask at least what things we want to say uncontroversially exist.

Now, as it turns out, basically nothing uncontroversially exists, but I think we can make a list of things we agree exist.

[crisp basalt]

glad we could lift this thread from the depths of "failure of measurement" math into the heights of "what is existence" philosophy

[tired kettle]

The only things whose existence I accept axiomatically are the real numbers. If they're not real, then that's something of a misnomer, and that is something I can not abide.

[prime glade]

Of course, whether [nothing] exists (no, not "ZFN") is highly controversial.

[prime glade]

I understand completely. Luckily, your first-order theory of the world has a model completely in the real numbers.

[tired kettle]

Well that's awfully convenient.

[obtuse silo]

I only accept the existence of totally separated spaces with > 3 points

[prime glade]

So, I want to say

• Shoes
• People
• Dogs
• Trees
• Photons
• Stars
• Pear
• Force fields

all exist.

Of the things I want to say do not exist,

• Sherlock Holmes
• Fictional characters more generally
• Causally inert objects
• God
• Supernatural forces

are some.

[tired kettle]

I would say that shoes, people, dogs, trees and myself are largely uncontroversial. I will take issue with stars, photons and force fields.

[tired kettle]

And "supernatural forces" is such a fucking copout for something nonexistent. If it existed we would by definition claim that it is natural and not supernatural.

[prime glade]

Haha, yes, in part this is true! Perhaps naturalism is trivial...?

[obtuse silo]

The present king of france, too.

[prime glade]

But I mean moreso stuff like what we lump in with "supernatural."

[prime glade]

Haha! Indeed!

Is he bald?

Do I appear jolly, everyone?

[prime glade]

To elaborate on this a bit, does anyone think there are objects which have no proper parts?

[obtuse silo]

The singleton

🥶

Well

[prime glade]

Alright, are you actually committed to the existence of the singleton or are you merely jesting?

[obtuse silo]

does the empty set count as a part

[prime glade]

Excellent question!

I would say so.

But there is a more general question when we talk about objects in general.

[obtuse silo]

I am commited to the existence of the extension of any concept(unless it entails a contradiction ala Russell’s paradox)

[prime glade]

If the empty set is the part of the singleton, it is presumably part of every thing.

[obtuse silo]

So just whatever works, honestly.

[prime glade]

What about the concept which causes all things to be true, like literally makes every statement true?

[obtuse silo]

🤔

[tired kettle]

What is an object and what is a part?

[obtuse silo]

Nah, no set of all sets.

[obtuse silo]

After the edit, I’m not sure what you mean.

[prime glade]

Uh, so by existing, it automagically turns everything true.

Like, every proposition.

[obtuse silo]

I’m not entirely sure we mean the same thing by “concept”

[prime glade]

Well, imagine I have the concept of an object C which I concieve of as "C is defined as any object which, by existing, causes all propositions to be true."

[prime glade]

An object is a thing, and a part is an object, ostensibly, within in some sense some object. For example, your arm is part of your body.

[obtuse silo]

By “concept”, I mean a function which is well-defined on all objects, and always outputs a truth value

E.g., ala Frege.

[prime glade]

Ah, I see.

[tired kettle]

This feels eerily similar to that, "God is the greatest possible thing; the greatest possible thing having the property of existence is better than the greatest possible thing having the property of nonexistence, so therefore, God must exist" argument used by that 12th century theology guy.

[prime glade]

Indeed!

[prime glade]

Well, then it's kind of vacuously true! In the same way "every thing exists" is.

[tired kettle]

I very much dispute this definition. I would argue that my laptop is an object; and yet it is not the icelandic parliament.

[prime glade]

This joke went over my head.

[tired kettle]

https://en.wikipedia.org/wiki/Thing_(assembly)?useskin=vector

In Iceland they still call their parliament "Thing".

[prime glade]

Pear, you should read Constructive functional analysis. I have yet to even open it nor download a PDF but it sounds awesome and is by my GOAT Bridges.

[prime glade]

Awesome.

[obtuse silo]

Read T&G first ✍️

[prime glade]

i can imagine how musty a physical copy of this book must be

so this book is like ancient

uh lots of stuff has happened in the last 50 years

apparently he ONLY considers metric spaces

i guess that makes sense but i was hoping for some basic acknowledgement of constructive topology but whatever

[rough rapids]

out of curiousity, how many hours do you estimate it would take you to read a book like this

[white ingot]

Cold war ahh book

[rough rapids]

ive actually never read a full textbook from start to finish, despite doing some work from many of them

[prime glade]

i think most people with some solid analysis background could get through this book in 2-3 weeks if they work diligently, it's more a monograph it's not like it has exercises or anything

[white ingot]

I don't usually read a book from start to finish. I just usually flip forward to an interesting topic.

[prime glade]

he proves the stone-weierstrass like 90 pages in

[rough rapids]

i like to start with problems and then flick my way back through until i can solve them

[prime glade]

he also develops only integration on locally compact spaces which makes sense for functional analysis

actually though i think pear would enjoy Techniques of Constructive Analysis, almost all the content in this book is subsumed there

[white ingot]

I consider the textbook cooler than the math class it teaches

[prime glade]

look at this

[white ingot]

o-o

That the proof?

[tired kettle]

Time to get the texromancers back together.

[prime glade]

okay so he's... hmm... im not sure how to feel about this

[white ingot]

I'm digging through my alg 2 textbook my teacher gave to never use, and I found a lot of interesting stuff

[prime glade]

good keynes this is his replacement

brooooooooooooo

[white ingot]

what?

[prime glade]

his replacement of the pretty fundamental result that all finite dimensional subpaces of a linear normed space are best approximation spaces is pure trash

this is his definition of "at most one best approximation" by the way

[ember marlin]

It commits you to some form of existence of Sherlock Holmes. Obviously Sherlock Holmes doesn't exist as a human person, but he does exist as a character with an established appearance, which is what gives meaning to the claim that someone looks like him.

[prime glade]

So, is there anything which does not exist?

And also perhaps sapphire can say the same thing.

[ember marlin]

A married bachelor doesn't exist.

A qwalrexivor doesn't exist.

[prime glade]

How can you talk about one, then?

To be clear, if you can say this, surely it is not a far stretch to say "x approximates this thing" where that thing doesn’t exist.

[ember marlin]

I mean, about the only thing I can say about it is that it doesn't exist.

[prime glade]

haha, surely you can also infer a married bachelor is married, and a bachelor, and a variety of other facts

presumably many statements that apply to married people apply to married bachelors

But regardless! It’s still worrying!

Presumably "married bachelor" refers to something lest "there are no married bachelors" be meaningless.

[obtuse silo]

Cleary just a disguised form of “there is no x such that married(x) and bachelor(x)”.

✍️

Or, equivalently, for every x, not (married(x) and bachelor(x))

[devout stag]

yes! why did they stop smh

i was recently looking into ways we can use ai here and it is looking not very bad

if i get some time to work on it you could have auto-tex-er

[prime glade]

Alright, based. Now, what about "all married bachelors are married?"

[obtuse silo]

That’s just “for every x, (married(x) and bachelor(x)) implies married(x)” ✍️ which is vacuously true

[prime glade]

Alright, so I presume you have the same account for "if X is a married bachelor, he is married?"

[obtuse silo]

Yeah

Although, a paraconsistent logic would probably be better for this than one that affirms the principle of explosion, if you want things like “married bachelors are married” to be true but not “married bachelors are green”

[prime glade]

Yup!

Anyhow, counterfactuals with impossible antecedents occur all the time, even in mathematics, and we don’t really want them to be trivial / vacuous.

[obtuse silo]

But that isn’t exactly related to discussing non-existent objects(after all, if it is true there are no married bachelors, every married bachelor satisfies P for any predicate P)

[prime glade]

There’s also the problem of capturing in virtue of what is any implication vacuous.

[prime glade]

Well, married bachelors would not exist, but there are married bachelors.

[obtuse silo]

True

✍️

[prime glade]

Existence becomes a predicate usually!!

And \exists becomes something like "there is."

[prime glade]

Kant is seething and coping right now

[obtuse silo]

Frege’s notion of existence was better fr.

Not a first order property, but a second order one ✍️

Same as being instantiated

[prime glade]

Frege was a GOAT, fr—I think given you’ve read Frege first you’ll stick near and dear to his tradition.

Whereas I’m very, very Quinean. Strangely, I like a ton of nonclassical stuff too.

[obtuse silo]

Ngl I didn’t just read Frege first, I’ve never read any other philosopher(aside from a little bit of unhinged Brouwer on women of all things)

[prime glade]

Ah, you probably should!

[obtuse silo]

I might read Kant, cause according to the internet Frege was a transcendental idealist before 1891, but I’m not sure what that even means entirely

[prime glade]

I do recommend Quine, he’s a very good thinker.

[prime glade]

For example, "if the halting problem were decidable, then we would be able to solve arbitrary Diophantine equations."

[tired kettle]

It was being run when a bunch of my friends were in undergrad and had a ton of time to kill; especially over the summer. (I had a job that summer, and so could not participate). Right now, most of the texromancers crew are in grad school.

[devout stag]

can we contribute??

let us young'ns do it

not like we have anything better to do

[tired kettle]

I'm not in charge of it. I've been thinking about trying to revive it, but I haven't had the time. If you want to do something, reach out to Aareyan Manzoor at Waterloo.

[cerulean junco]

The circumference and diameter are never both rational

[ember marlin]

thr circumference or a diameter of a circle must be rational i.e. have a terminating decimal expansion because they are definite
Why must a number be rational to be definite?

[craggy delta]

use the madhava-lebinz series

$lim _ {p \to \infty} \sum_{n=1}^p\frac{(-1)^{n+1}}{2n+1}$

[vast spear]

curry supplier

[merry lava]

k not p ?

[craggy delta]

mb

[white ingot]

Ok so how is e irrational if its series is based upon the sum of reciprocal factorials?

[crisp basalt]

infinite sequence of rational numbers can converge to an irrational

(the sequence here being the sequence of partial sums, each of which is rational)

[ember marlin]

In fact, that's basically trivial. For any irrational, you can construct a sequence of rational numbers that converges to it by just making the nth term of the sequence equal to the irrational to the nth decimal.

[hidden jay]

There is an exercice I know that uses this fact you set s_n=1+1/2+..+1/n! and consider v_n=s_n+1/(nn!) then you can prove that (s_n) and (v_n) are adjacent and have limit e, in that case for all n you have s_n<e<v_n if e were rational you would get a contradiction because if e=p/q, p>0,q>0, being integers then you would have q!qs_q<q!p<q!qs_q+1, you would have an integer contained between two consecutive integers strictly (not possible)

[craggy delta]

$\underset{k \to \infty}{lim} \sum{n=1}^p\frac{(-1)^{n+1}}{2n+1}$

[vast spear]

curry supplier

[hidden jay]

You can write it this way $\underset{k \to \infty}{lim}$ if you prefer

[vast spear]

R

[craggy delta]

ohhh

ty

$\underset{k \to \infty}{lim} \sum{n=1}^p\frac{(-1)^{n+1}}{2n+1}$

[vast spear]

curry supplier

[tawdry swift]

💀

$\lim_{p\to\infty}\sum_{n=1}^p\frac{(-1)^{n+1}}{2n+1}$

[vast spear]

Kocher

[rain heath]

This has been going on for so long

Can someone tl;dr this because I want to join

[tawdry swift]

Someone decided to join in (mind you, not OP!) and basically start a really long argument on Euclidean ideologies and how basic shit is defined

There’s so many topics discussed in this thread that you can’t summarize it properly

[tawdry swift]

Ain’t this just pi/4 though

Or rather -pi/4

Nah not really

$\arctan(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{2n+1}$

[vast spear]

Kocher

[tawdry swift]

No I’m right lol

It’s -pi/4

,w sum from n=1 to infinity (-1)^(n+1)/(2n+1)

[vast spear]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by exiled_hype

[tawdry swift]

Oh minus 1

Yeye

[rain heath]

I was reading for around 5-10 minutes about "irrationality of pi proof but simplified" and i was like "why does this keep going?!"

[supple spire]

We had the same question

"why did this keep going?"

[merry lava]

don't forget your _s lol

[craggy delta]

ye rhat mb gng

[gusty burrow]

Because irrationals like Pi do, they do not land on any fixed quantity comprised of countable quanta.

[supple spire]

This ping genuinely scared me

[ember marlin]

I've literally directly explained to you how they do.

[gusty burrow]

No, you have been refuted at every point. I have clearly explained my position as to how they do not.

[ember marlin]

No you haven't.

[gusty burrow]

How do they land, on a fixed quantity comprised of countable quanta (or elementary units)?

That is right, they do not.

Unless you make them rational.

[ember marlin]

P1. Any set which can be placed in bijective correspondence with any subset of the natural numbers is countable.
P2. The set of all natural numbers is a subset of the natural numbers.
P3. The digits of any finite real number can be placed in bijective correspondence with the natural numbers.
Cl. The digits of any finite real number are countable.```

[gusty burrow]

That does not answer the question.

How does an infinite convergent sequence, produce a specific outcome/result/quantity that is comprised of specific countable elements or quanta?

[ember marlin]

Yes. It does. It exactly does. A finite real number is fully described by its digits, which are countable.

[gusty burrow]

For an irrational, the digits are not all countable, they have no specific quantity or measure.

[ember marlin]

...yes, the digits of any finite real number are countable.

[gusty burrow]

Although the digits of any real number can be indexed by the natural numbers (and are therefore “countable” only in the sense of correspondence, this is not my point and you know this already), this indexing merely arranges an infinite, convergent process.
It does not provide a finite, fully constructed object, a requirement for genuine “numberhood”. The infinite list remains an idealized procedure rather than a completed, measurable entity.

Although we can, in principle, assign a natural number to each digit of π (first, second, third, and so on), you can never finish counting them all because π has infinitely many digits. Thus they are not a specific quantity comprised of countable elemental units or quanta.

You response does not bypass the central issue I have to point out again and again to you: the infinite tail of the expansion is never finitely “landed upon.” The claim that a finite real number is completely described by its countable digits glosses over the fact that the process of engaging with infinitely many digits is inherently non-finite. In essence, their totality embodies an infinite process that falls short of constituting a fully explicit, finite construction. A point which is paramount to my critique of irrational numbers, you repeatedly avoid addressing or acknowledging.

[ember marlin]

If you don't mean "countable", then don't say "countable".

[gusty burrow]

Specific quantity, comprised of countable elemental units or quanta.

Where is your specific quantity?

Comprised of a specific amount of quanta?

You have none.

As always.

[ember marlin]

Right, "countable". Like the digits of any finite real number.

Or the set of all rational numbers less than a given finite real number, i.e. the Dedekind cut defining the number.

[crisp basalt]

it does make me uncomfortable that we define the set of real numbers, while at the same time most particular real numbers are not definable

(there are pointwise definable models of set theory...)

(except in those models R is much smaller than we actually want, they are actually countable but the model does not know how to count them)

[ember marlin]

What numbers aren't definable?

[crisp basalt]

most reals

y'know, there are uncountably many reals but countably many finite texts

π is definable, we can define it as the algorithm that produces its digits or whatever (this is actually kind of a funny way to think about going about things, you can define the sum of 2 algorithms as the algorithm that runs the first two and adds the digits of their progressively more accurate outputs), but most reals aren't

[gusty burrow]

That does not make an infinite sequence, produce a realized finite specific outcome or quantity comprised of countable elemental units. They are not realized specific contructible quantities.

Dedekind cuts are not proof of measurement or common measure, they are axiomatic abstractions.

[ember marlin]

You're talking about computability, which is arguably different than definability.

[crisp basalt]

i really want to understand rxrsapphire, their position seems apparently defensible and believable, but everything they say just looks like nonsense

[crisp basalt]

yeah it is, but you can't really define definability

so i just used computability instead

[ember marlin]

What the fuck is an "elemental unit"? Because if it's a digit, then shut the fuck up because every finite real number has countably many digits.

[gusty burrow]

What do you want me to explain further?

[crisp basalt]

and it doesn't change the fact that there are Z many finite texts

[gusty burrow]

No, they (the irrationals) do not have a specific quantity of countable digits.

[crisp basalt]

most of R is somewhat unknowable, which is very strange
i have heard the phrase "finding hay in a haystack" used to describe this

[ember marlin]

...yes, they do. They have countably infinitely many digits.

[crisp basalt]

i have also heard the expression "where is all the cheese hiding" used to describe this
https://www.youtube.com/watch?v=aYuzwNa0_4o

which is very funny

[gusty burrow]

lol

[ember marlin]

What?

[gusty burrow]

Still waiting on that specific quantity.

[ember marlin]

...infinity. Countable infinity. Aleph null.

[gusty burrow]

Comprised of a specific amount of countable quanta or elemental units in some representation.

[gusty burrow]

No such thing.

[ember marlin]

Yes, there is. It's the cardinality of the set of natural numbers.

[crisp basalt]

no such set!!

[gusty burrow]

It is not the specific quantity, comprised of a specific amount of countable quanta or elemental units in some representation, that I keep asking you for.

[ember marlin]

Maybe you should use words that mean what you're trying to say instead of words that either mean something completely different or don't mean anything at all.

[crisp basalt]

from what i have heard, she only believes in positive integers, basically
rationals only exist as ratios of integers, not numbers in themselves

[gusty burrow]

You can represent them in some form as countable elements.

ie 1/3 is # relative to # # #

You cannot do that with irrationals. Thus they have no specific quantity, comprised of quanta.

[crisp basalt]

i just really like the idea that you could take a cheese and cut it with a thin knife at every possible computable position, and the cheese would remain 100% there

[ember marlin]

USE. WORDS. CORRECTLY.

[crisp basalt]

this is "real cheese", not to be confused with real cheese, which is made of atoms

[ember marlin]

...I mean, the cheese would "remain 100% there" regardless, as cutting cheese does not remove any of it.

[crisp basalt]

if you cut it at every possible real position, then it would disappear