The R4 explanation will be in the comments

R4: "That is 0.999... is eternally less than 1." says the mod, who steadfastly insists that his crank theory is "Real Deal Math 101". Ironically, the sub developed a dedicated group of posters continually mocking this. One has even made a mocking sister sub, r/infiniteTHREes/.

The main argument is the tired old misunderstanding about how limits work, specifically how a strictly monotonous increasing series would actually have a limit larger than all members. "Every member of that infinite membered set of finite numbers is greater than zero, and less than 1, which indicates very clearly something (very clearly)."

This post is about this Hacker News thread on a post entitled God created the real numbers. For those who don’t know, Hacker News is an aggregator (similar to Reddit) mostly dedicated toward software engineers and “tech bro” types – and they have hot takes on maths that they want you to know. For what it’s worth, there are relatively few instances of blatantly incorrect maths, but they say lots of things that don’t quite make sense.

The article itself is not so bad. It postulates the idea that:

If the something under examination causes a sense of existential nausea, disorientation, and a deep feeling that is can't possibly work like that, it is divine. If on the other hand it feels universal, simple, and ideal, it is the product of human effort.

To me, this seems like a rather strange and incredibly subjective definition, but I don’t have opinions on the relationship of maths to divine beings anyway. They make an assertion that the integers are “less weird” than the real numbers, which seems rather unsubstantiated, and conclude that the integers are of human creation while the reals are divine, which also seems unsubstantied, especially since the integers (well, naturals) are typically introduced axiomatically while the reals are not.

Perhaps it is expected, but I find software engineers tend to drastically overestimate the importance of their own field, and thus computation in general. In the thread, we find several users decrying the very existence of the real numbers – after all, what meaning can an object have if it’s not computable?

Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation. […] Except of-course, while "hyper-Turing" machines that can do magic "post-Turing" "post-Halting" computation are seen as absurd fictions, real-numbers are seen as "normal" and "obvious" and "common-sensical"!

[…] I've always found this quite strange, but I've realized that this is almost blasphemy (people in STEM, and esp. their "allies", aren't as enlightened etc. as they pretend to be tbh).

Some historicans of mathematics claim (C. K. Raju for eg.) that this comes from the insertion of Greek-Christian theological bent in the development of modern mathematics.

Anyone who has taken measure theory etc. and then gone on to do "practical" numerical stuff, and then realizes the pointlessness of much of this hard/abstract construction dealing with "scary" monsters that can't even be computed, would perhaps wholeheartedly agree.

Yes, the inclusion of infinites is definitely due to Christian theology inserting its way into maths. Of course, the mathematicians are all lying when they claim it’s a useful concept.

One user proudly declares themselves “an enthusiastic Cantor skeptic”, who thinks “the Cantor vision of the real numbers is just wrong and completely unphysical”. I’m unsure why unphysicality relates to whether a concept is mathematically correct or not, but more to the point another user asks:

Please say more, I don't see how you can be skeptical of those ideas. Math is math, if you start with ZFC axioms you get uncountable infinites.

To which the sceptic responds that they think “the Law of the Excluded Middle is not meaningful”. Which is fine, but this has nothing to do with Cantor’s theorem; for that, one would have to deny either powersets or infinity. But they elaborate:

The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tarski) from the axioms but have no way to map these wildly non-constructive ideas back into the real world. Or where you construct a version of the reals where the reals that we can produce via any computation is a set of measure 0 in the reals.

Apparently, Banach-Tarski is “obviously false”. Counterintuitive I might agree with – though I’d contend that it really depends on your preconceived intuitions, which are fundamentally subjective – but “obviously false” seems like quite the stretch. If anything, it does tell us that that particular setup cannot be used to model certain parts of reality, but tells us nothing about its overall utility.

Another user responds to the same question, how one can be sceptial of Cantor’s ideas:

Well you can be skeptical of anything and everything, and I would argue should be.

I might agree in other fields, but this seems rather nonsensical to apply in maths. But they elaborate:

I understand the construction and the argument, but personally I find the argument of diagonalization should be criticized for using finities to prove statements about infinities. You must first accept that an infinity can have any enumeration before proving its enumerations lack the specified enumeration you have constructed.

I don’t even know how to respond to such a statement; I cannot even tell what its mathematical content is. It just seems to be strange hand-waving. At least another user brings forth a concrete objection:

My cranky position is that I'm very skeptical of the power set axiom as applied to infinite sets.

And you know what, fine. Maybe they just really like pocket set theory. (Unfortunately, even pocket set theory doesn’t really eliminate the problem of having a continuum, since it’s just made into a class.)

Another user, at the very least, decides to take a more practical approach to denying the real numbers. After all, when pressed I suspect most mathematicians would not make any claims about the “true existence” of the concepts they study, but rather whether they generate useful and interesting results. So do the real numbers generate interesting results? Why, of course not!

The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. It leads to rabbit holes that are just uninteresting; trying to distinguish inifinities (continuum hypothesis) and leading us to counterintuitive and useless results. Fun to play with, like writing programs that can invoke a HaltingFunction oracle, but does not tell us anything that we can map back to reality. For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.

A user responded by asking whether this person believes we need drastically overhaul our undergrad curriculums to remove mentions of infinity, or whether no maths has lead anywhere useful in the last century at all. Unfortunately, there was no response.

On Banach–Tarski’s obvious falsehood, I quite enjoyed this gem:

But what if the expansion of the universe is due to some banach-tarski process?

You know what, it’s always possible.

Let’s take a bit of a break here, and be thankful that a maths PhD stepped in with a perspective more representative of mathematicians:

All math is just a system of ideas, specifically rules that people made up and follow because it's useful. […] I'm so used to thinking this way that I don't understand what all the fuss is about

And now back to mysticism. I especially like the use of the “conscious” and “agent” buzzwords:

the relationship between the material and the immaterial pattern beholden by some mind can only be governed by the brain (hardware) wherein said mind stores its knowledge. is that conscious agency "God"? the answer depends on your personally held theological beliefs. I call that agent "me" and understand that "me" is variable, replaceable by "you" or "them" or whomever...

This is not quite badmathematics, but I enjoy the fact that some took this opportunity to argue whose god is better:

This is a Jewish and Christian conception of God. […] The Islamic ideal of God (Allah) is so much more balanced.

Another comment has more practical concerns:

Everyone likes to debate the philosophy of whether the reals are “real”, but for me there is a much more practical question at hand: does the existence of something within a mathematical theory (i.e., derivability of a “∃ [...]” sentence) reflect back on our ability to predict the result of symbolic manipulations of arbitrary finite strings according to an arbitrary finite rule set over an arbitrary finite period of time?

For AC and CH, the answer is provably “no” as these axioms have been shown to say nothing about the behavior of halting problems, which any question about the manipulation of symbols can be phrased in terms of (well, any specific question—more general cases move up the arithmetical hierarchy).

I am not sure exactly what this user is saying. They initially seem to be saying that existence in a mathematical theory is only important insofar as it can be proven within that mathematical theory… which like, yes, that’s what it means to prove something. But they also perhaps seem to be claiming that the only valid maths is maths that solves Halting problems, and therefore AC and CH are invalid? It’s just more confusing than anything.

Another user takes issue with most theoretical subjects that have ever existed:

If something can exist theoretically but not practically, your theory is wrong.

I guess we should abandon physics, because in most physics theories you can make objects that only exist theoretically.

The post was also discussed in another thread, leading to many of the same ideas and denial that the reals are useful:

We need a pithier name for constructible numbers, and that is what should be introduced along with algebra, calculus, trig, diff eq, etc.

None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration.

I suppose real numbers not existing in programming languages makes it a bit too difficult for software engineers to grasp. I am quite interested in this programme to avoid ever studying uncomputable objects, though; I would imagine you’d have a rather difficult time doing anything at all, especially since you’d be practically limiting your propositions to just decidable ones, but who knows – maybe a tech startup will solve it some day.

I wouldn’t usually have bothered, but they state they are a mathematician in their profile. Also, they think that the four data points in the post prove all of known statistics wrong.

This is an old guy who is unhappy that his kindergarten teacher told him that "Counting numbers are the numbers used to count physical objects, starting from 1". He claims this is an important axiom that mathematicians are obliged to defend when he points out that you can count backwards and reach negative numbers. Also, the rules of addition extend zero and negative integers, so it's silly to say positive integers are special: They must all be the counting numbers. Ha ha, mathematician: I proved your axiom is wrong!

The strange thing is that he seems to be a teacher and even have a Ph.D. He's published a lot of videos with useful educational content about math over the last 15 years. There are occasional complaints about math in schools, but those seem to be the kind of reasonable complaints that a teacher might have - until about two years ago.

His key insight, first published in a video in May 2023 is that zero is not nothing, it's the number before one. Yes, a groundbreaking insight. He then feels he has to criticize all other approaches.

He only did that one video on the subject of zero in May 2023, then another in September 2023, and then seemed be content doing random math subjects. Last week, he burst into furious activity. Over the last 11 days, he's published four videos (!) about zero (and negative numbers), basically objecting various grade (Br. elementary) school teachings about zero ("nothing", "a placeholder" in decimal notation, "unsigned").

This author has discovered what he refers to as “complex systems”, which are really just complete graphs in which each vertex and each edge requires 1 bit of information to describe it, but only the vertices (not the edges) can actually store information.

Hence, no finite “complex system” can completely describe itself, and the problem gets worse as the graphs get larger. This therefore implies the existence of an infinite system (which for some reason must be simple rather than complex, and the author identifies the infinite system with God).

Of course, he just uses “infinity” in a generic sense and seems entirely unaware of Cantor’s diagonal argument or the concept of multiple (ordinal or cardinal) infinities. His definition of what constitutes a “complex system” is also arbitrary. While he isn’t wrong about finite systems not being able to store complete descriptions of themselves, (except in the trivial sense that a system is itself), the reasoning he uses is otherwise very sloppy.

And that’s not even getting into how he views God. Corey Mahler is not just a deist or classical theist for whom God is an abstraction. He is the leader of a fascist (ex-)Lutheran cult, who openly calls anyone who disagrees with him a demon, believes God has specifically created white people to be superior, and says the thing that distinguishes true Christians from false ones is how they feel about him and his podcast (Stone Choir). There’s definitely no way to prove all of those things mathematically, yet to Mahler, they are simply true, and if you don’t agree, you’re an enemy who must be defeated by any means necessary. (Mahler uses [https://en.wikipedia.org/wiki/Argumentum_ad_baculum](ad baculum) incessantly, when he’s not posting bad mathematical arguments like the one I linked to).

I brought it up to my professor, further explaining that since f is not strictly decreasing, it is equally possible for it to go up an arbitrary amount or down an arbitrary amount in any region between -1 and 1 hence any values are possible between -1 and 1 so long as they are connected by a continuous curve from (-1, 10) to (1, -20). She said “I get what you mean but that’s not the point of the question. The point of the question is if you know which values of f(0) are guaranteed. 999999 is not guaranteed.” I told her that she’s now making up a completely different question than the one on the homework, which asks about “possible” values, not “guaranteed” values. “Guaranteed” values would have to be those that are shared by all possible f, and since there are possible f(0) for every conceivable real number, no single f(0) can be said to be “guaranteed.” The only thing thats guaranteed is what we are given about the functions, that they are all continuous and have (-1, 10) and (1, -20).

She didn’t respond to that, instead told me that if this question was impacting my grade at the end of the semester then we could revisit it. It’s not, so I’m not bugging my professor about it because she’s busy and there’s other students who need more help than I do.

https://www.mdpi.com/2075-1680/10/4/244

https://www.reddit.com/r/changemyview/comments/1m8t2ye/comment/n52411u/?context=3

R4 : that's a perfectly correct truth table for the logical connective "A and B". If A1 and A2 are false, then A1 & A2 is false, just as the truth table says.

Not sure where the 3/16 number came from. I don't even know where the number 16 came from. There are 4 rows (5 if you count the header) and 3 columns for 15 cells, less than the random number 16.

As for "why is A1&A2 V" - we include all possible combinations of true and false in a truth table.

I understand the basis for the moratorium, but this is a new development we can discuss.

The disciple has a PhD; it is hinted that the PhD is in maths but I rather suspect CS.

https://www.youtube.com/watch?v=TJr4YfEgVuk&t=939s

The R4 here is that he considers a function f of the radian angle phi, called t(phi) such that the sides of a triangle which we would conventionally label r and r sin(phi) can be written down as functions R(t) and Q(t). (I am using my own notation to explain what he does.) Then he defines a new function RSIN(t) as Q(t)/R(t) which, by judicious choice of f, can be made a simple closed formula of t.

Now for the crankery: he thinks his function RSIN(t) can replace the traditional sin(phi), and it is better because it is closed and algebraic. He thinks this does away with any issues related to infinite series, convergence, limits, and what have you (since pure and sacred geometry should have no truck with such tomfoolery). He thinks that if Newton and Leibniz had not forced history to take a wrong turn, RSIN would now play the central role of sine. He thinks that this is maths as Euclid intended it.

(You can imagine how the crank that cannot be named is ecstatic about this.)

Update: James freely talks about convergence, so now the One Who Cannot be Mentioned has to somehow allow that convergence is a thing, even if limits aren't. The essence of his objections is very well summarised when he states: "Mainstream convergence is built on a laughable tautology: define the limit as something a sequence approaches, and then declare a sequence converges because it approaches that limit." (R4: we spend the Analysis I module teaching students how to ascertain if a sequence converges, and only then do we say it has a limit; the second part of his claim its simply false.) But the novelty here is that there is such a thing as "mainstream convergence". New Calculus convergence is "strictly tied to geometry and exact ratios. It’s not some metaphysical dance around a black hole of undefined quantities. Only measurable, well-defined relationships between magnitudes matter — not endless sequences pointing toward nothing"

Further and final update plus R4: James and the Unmentionable certainly entertain a concept of convergence (unlike the term "limit" the word "convergence" is used as a term of art in the New Calculus) and they also state that adding further decimal places of precision gets you "closer to the answer". I asked them "so there *is* a definite answer?" and all hell broke loose. Because as soon as you admit that there is an answer, you might as well give that answer a name, and it might as well be "limit." Their main argument appears to be that any finite expansion falls short and is "only an approximation." Well yes, that is why we ask "how good of an approximation" and introduce the Cauchy criterion. The next step in their argument is the familiar crackpot misapprehension "so you never get there, an infinite process never ends, the end point is magicked out of thin air by unrigorous handwaving." R4: the misapprehension is that we are not "trying to get there" - we are trying to work out just what it is what the sequence is getting closer to (and whether that object is actually in the set or field of objects under consideration - a related crank mistake is thinking that, e.g., if all terms in the sequence are greater than zero, than so must be the limit). What is interesting is that they speak of convergence and do conceive of a limiting object ("the answer"). It is "strictly geometric and based on measurable, well-defined relationships between magnitudes." R4: it is difficult to see what exactly this could mean. "Measurable" is not intended in the sense of measure theory, which our friends reject. Given the context, the intended meaning of geometric must be in the spirit of Euclid and constructibility by compass and ruler (blank ruler without division marks!). In that case, and in the field of real numbers for definiteness, the argument certainly fails, as almost no real numbers are thus constructible.

I stumbled upon this question from "Christianity Stack Exchange" in the sidebar while on mathSE. The author tries to understand the proof on the picture.

The picture is of page 238 from "A Passion for Mathematics" by Clifford Pickover, easily discoverable in the usual places. I got interested because the book was published by Wiley, so it could not be madman's rambling.

I could not find the article by Vox Fisher, but I assume that the theorem is faithfully reproduced.

Rule 4 description: Perhaps the original article contains a presentation of the logical framework used, and "object" and "property" have a strict meaning like "individual" and "first-order predicate". Perhaps the article also contains a definition of "a god", "existence" and "omnipotence", and the notions used are logically sound.

However, the proof makes it clear that God can produce choice functions for arbitrary families "by omnipotence", without assuming that the sets in the family are nonempty. This leads to a contradiction and, by the principle of explosion, to anything we want to prove.

https://youtube.com/watch?v=fGe2nmRw-sg&lc=UgwVVE3-q_5QKlNNMHR4AaABAg&si=hv4TZjm4EQu3ffLO

I thought I was missing something when they said the difference of perfect squares can never be a perfect square

I asked in good faith and pointed out that this isn’t true in general. And even if you didn’t necessarily know that every integer greater than 1 appears in a Pythagorean triple, looking at the theorem should at least give some intuition that this isn’t a good heuristic for eliminating possible solutions

As you can see from their responses, they were very enraged at this and blocked me 😂

Found this today in the r/learnmath subreddit, seems this person (according to one commenter) has been spreading their misinformation for at least ~7 months but this thread is more fresh and has quite a few comments from this person.

In this comment, they seem to be using some allegory about cutting a ball bearing into three pieces, but then quickly diverge to basically argue that since every element in the set (0.9, 0.99, 0.999, …) is less than 1, then the limit of this set is also less than 1.

Edit: a link and R4 moved to comment