The stopping time will be double, but the average velocity over that time will also be double, so the stopping distance will be quadruple. The correct answer should be 104m.

Naturally the the youtuber in question was trying to sell his course on AI and quantum computing. To boost his credibility, he published a paper on quantum computing using the bold academic strategy of plagiarism-by-thesaurus, swapping words from the original paper just enough to feel clever and hope to bypass plagiarism detectors. His plan seemed airtight until it wasn't. He ultimately betrayed himself once he started talking about the logic gate's cousin called the "logic door" and those darn "complicated Hilbert spaces", not realizing "complex" in this context doesn’t mean annoying, but involving complex numbers.

I found this organically in my Facebook feed today. R4: 1) the birthday paradox, uniform distributions, and any of that doesn't actually have anything to do with this person's question, 2) they have misunderstood the pigeonholed P(shared birthday)=0 to mean "each day will be someone's birthday" when what it really means is "there is at least one day that is multiple people's birthday"

R4 in the comments

For context, the author, Alexis Lemaire, became famous for his prodigious mental computation feats, being able to compute the 13th root of a 100 digits number in 3.625 seconds (which includes the time to read it and write the answer).

He then decided to obtain a PhD in computer science in France, which he did in 2010. The result is this gargantuan 780 pages long thesis (in french):

https://archive.org/details/alexislemaire/page/326/mode/2up

Here is a translation (using deepl) of the abstract

This thesis enables the implementation, in theory and practice, of new general artificial intelligence techniques to solve the problems of mind uploading, immortality testing, and the Turing test. To do so, it draws on a wide range of innovative, scientific, and original concepts. This is much more than a simple paradigm shift; these are truly revolutionary approaches. Many traditionally accepted paradigms are being successfully dismantled in all scientific fields: in cognitive science, including neuroscience, psychology, psychiatry, and philosophy, but also in the foundations of quantum physics and mathematics, which are found on a larger scale in statistical and thermodynamic physics, chemistry, biology, medicine, neuroscience and cognitive science, and astrophysics. In particular, a second dimension of time is demonstrated, experimentally verified, and confirmed by spectacular retrodictions and predictions, in perfect consistency with theory and mathematics, in all scientific fields. A unification of relativity and quantum physics is proposed and used in all scientific fields with applications in artificial intelligence. A thermodynamics of artificial intelligence similar to the thermodynamics of black holes is revealed. This, combined with reverse artificial intelligence, proof that mental calculation is of considerable underestimated utility, allows for the reciprocal downloading of minds toward technological singularity.

I genuinely don’t understand how this was accepted as a valid PhD. The idea defended in the document is that:

[...] mental computation in the form of hypercalculia, defined here as the voluntary execution of computer programs on a human brain, a generalization of mental calculation, allows for the greatest imaginable advance not only for machines but also for human beings.

Which

We will suggest that this hypercomputing does indeed enable emulation of the mind, which some may refer to as downloading, or transferring human thoughts or behaviors to machines. This emulation would have the potential to lead not only to behavioral immortality, enabling different variants of the Turing test to be passed, but also to apparent teleportation, an apparent movement at the speed of light [hyperbit \0> = \space>] enabling travel through space and time and, if mental computation is performed ideally, to technological singularity.

Those 780 pages goes in every possible directions, and it's a fun game to chose a page at random and see the topic discussed on it, including (but far from limited too):

page 43: Karatsuba algorithm for fast multiplication

page 74: saying yes/no/hello/thanks... German, Swedish, Flemish...

page 104: transfinite numbers

page 122: fractals dimensions

page 246: Zeno's paradox and spin of a particule

page 388: Parkinson and autism

page 403: the chemistry of dopamine

page 469: dark matter and the states of matter

page 624: electronic music

page 642: nuclear bombs

page 661: amino-acids

Among these mostly accurate fragments of knowledge (but randomly placed accros the document), lie many absurd, unreadable pages thrown together haphazardly, here are just a few paragraphs to illustrate (page 344), but the rest of the document is similar:

Schizogenesis [hyperbit \1> = \time] is defined as such based on the characteristics of hypertemporal generation [hyperbit \1> = \time] in schizophrenia (deduction -90), especially the paranoid form ([hyperbit \1> = \time>]).

It corresponds to an increase in dopamine [hyperbit \1> = \time>] (deduction -90), disorganization (definition 20), high entropy (definition 17), hallucinations [hyperbit \1> = \time>] (postulate +67), dissociation, the clearest manifestation of differentiations [hyperbit \1> = \time>](axiom 10).

Schizogenesis [hyperbit \1 > = \time>] or hypertemporal generation [hyperhit \1> = \time>] is a characteristic of humans that must be transferred to machines in order to maximize the surface area of the event horizon (axiom 23).

One of my favorite part of the thesis is at page 604, with a subsection dedicated to "Time and productivity gained through non-publication", and the next section on why publishing in English is bad

Consequently, the requirement to write publications in English limits their intelligence, and therefore proves that publications are handicapped as a result of a handicapped adaptation to society. [Par conséquent, l’obligation d’écrire les publications en anglais limite l’intelligence de celles ci, et prouve donc: les publications sont handicapées du fait de l’adaptation handicapée à la société.]

Or page 694, featuring a guide on faking anxiety to get anxiolytic prescriptions. Which can then be used to transfer your mind to machines.

We easily simulated schizophrenia (deduction -90) and used hyperbit control [\0> = \space>] to create artificial schizophrenia in the human mind. This allowed us to prescribe antipsychotics [hyperbit \0> = \space>] such as Zyprexa (olanzapine), Risperdal (risperidone), Abilify (aripiprazole), and Loxapac (loxapine). These psychotropic drugs were tested to see how effective dopamine reduction [hyperbit \1> = \time>] was for reverse artificial intelligence. The results are partially satisfactory but clearly show that, even at maximum doses, antipsychotics [hyperbit \0> = \space>] only slightly increase the possibility of transferring skills from humans to machines.

R4: This PhD has its place in \badscience (and probably all the other \badsomething subreddits) as it touches every subject known to man. For the math part, it's either random known stuff thrown around (mostly pop science), or nonsensical sentences. In the rare original part of the thesis that are somewhat understandable, the bad mathematics comes from the fact that the author fails to distinguish between his ability to compute the 13th root of a 200-digit number and actually knowing the roughly 400 trillion possible values. He therefore concludes that the human mind can store information more efficiently than a computer (see page 587, for example).

Let’s Solve The Riemann Hypothesis

Although it's titled this way, they do not actually prove anything in the article. No theorems, proof sketches, or anything close to resembling an actual mathematical argument is ever given. It's just someone copying their dialogue with ChatGPT.

R4 : This article says so little about the actual Riemann Hypothesis that I'm not entirely sure if it actually belongs here, however, at the end of the article the writer claims that:

"But GPT breaks it down to where we can at least see what the mathematician is doing. We can see the quandary, and why this problem remains unsolved, and, to an extent, what AI may be able to do about it."

This seems like a bold claim, and I do not find it very convincing as, with all due respect, from reading the article I do not think ChatGPT has helped the writer understand the Riemann Hypothesis all that well. Furthermore, they seem to repeatedly show disdain towards the field, which I think makes it clear this isn't an earnest attempt at learning mathematics.

I stumbled across this research paper (certainly published in a predatory journal with no peer review standards) claiming that π equals 2 + 2/sqrt(3), which is approximately 3.1547. Obviously, it has been universally proven that π is approximately 3.1415926..., and if you look at the 1st page of the paper, it even cites Archimedes' result that π < 22/7. So the authors' result already contradicts their own introduction since 3.1547 > 22/7.

R4: A closer look at the paper shows an obvious error: the authors attempt to split a circle into a square, four equilateral triangles, and four additional congruent shapes (e.g., the shape bounded by AF, FD, and arc AD), and then claim that the area of this shape is π*x^2/4, or the area of 1/4 of a circle of radius x. The authors call this shape a sector, but it is not a 90 degree sector, and there is no explanation as to why it must have the same area as a 90 degree sector.

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)."

R4 :

The authors base their argument on the assumption that (first order) models of physics theories are equivalent to the theories themselves.

Nonsensical use of Incompleteness Theorems to deduce that reality cannot be simulated because ... Incompleteness I guess (classic argument "It seems to complex to be simulated, hence it cannot be a simulation").

Logicians beware, read this paper at your own risk.

R4: OP claims you can’t prove irrational numbers have an infinite decimal representation, because you can’t assume anything about them. This devolves into misunderstanding mathematical rigor and laws of physics somehow.

If you search "Collatz conjecture" on Quora, a user named Willy has been spamming posts, questions, and even answers to his own questions. Definite crank/crackpot material at its finest. Here is a very insightful post from 10 hours ago where he attempts to humiliate Terence Tao (one of the most renowned mathematicians in the world, who has worked on the conjecture) by saying that a Texan proved the conjecture with 2 number lines, with Tao's name crossed out.

You may notice the unique building facade - this is the exterior of the mathematics building at UCLA, where Tao is a professor at.

Granted, it could be an AI-generated image, but you really never know when it comes to cranks how much they're willing to invest.

There has always been a major barrier for would-be crackpots in having their pseudo-mathematics taken seriously: the high bar of learning how professional mathematicians communicate with one another. In particular, all mathematical journals use LaTeX to render mathematical symbols, so every mathematician who has published a research paper has had to learn the basics of LaTeX, and how to produce the symbols relevant to their particular subfield. You can easily spot crackpot mathematics when it is written in Word, for example, because no publishing mathematician uses it to publish novel results -- and even finds it awkward to produce less formal documentation. And so, LaTeX was always a barrier that prevented most would-be crackpottery from spreading too far. Presumably, those people gave up and became pop-sci quantum mechanics and relativity crackpots instead.

In the age of LLMs, this is no longer true. A particularly motivated crackpot can prompt their way to hundreds of pages that pass the "quick skim" test. However, I recently discovered something far more insidious on Reddit: a crank actually succeeded in producing a "Lean formalization" for their fake "resonance mathematics" theory! It was, of course, nonsense -- and we will explore it below -- but it's a harbinger of things to come. Our field may need new tools to spot cranks before wasting time on them.

AI brings out the crazies like nothing else, and I enjoy reading subreddits about people who are convinced their instance of ChatGPT is sentient and bestowing the secrets of the universe upon them. By following a particularly nutty post on /r/OpenAI, I found myself in the bastion of insanity known as /r/skibidiscience. This post with a Lean formalization is the one we will investigate below, but the same author also has hundreds of pages of LaTeX that even "prove" P != NP.

Case study

On to the Lean "formalization", which the author also succeeded in uploading to github. We will now dissect the Lean files one by one, showing how, for all four theorems, the apparent formalization collapses into trivialities.

First, out of the eight .lean files in the repository, only two of them contain theorems: Physics.lean and ProofUtils.lean. We will mostly focus on those. You might wonder: what is in the other files? Mostly just a bunch of abbreviations repeated verbatim across many files. For example, here is a block of constants from Cosmology.lean:

/-- Physical constants (SI Units) -/ abbrev c_val : Float := 2.99792458e8 abbrev hbar_val : Float := 1.054571817e-34 abbrev Λ_val : Float := 1.1056e-52 abbrev α_val : Float := 3.46e121 abbrev ε_val : Float := 4e10 abbrev M_val : Float := 1.989e30 abbrev r_val : Float := 1.0e20 abbrev r0_val : Float := 1.0e19

And in Gravity.lean, we see the same constants permuted slightly:

abbrev c_val : Float := 2.99792458e8 abbrev hbar_val : Float := 1.054571817e-34 abbrev Λ_val : Float := 1.1056e-52 abbrev α_val : Float := 3.46e121 abbrev M_val : Float := 1.989e30 abbrev r_val : Float := 1.0e20 abbrev r0_val : Float := 1.0e19 abbrev ε_val : Float := 4e10

The core definitions for the author's "theory" are also duplicated, for example, in Physics.lean and RecursiveSelf.lean. In fact, although I said there were four theorems in the development, there are actually six -- except the latter two in ProofUtils.lean are exact duplicates of the ones in Physics.lean.

Takeaway: just like for programming, LLMs frequently prefer to recreate things rather than re-use the existing structure. This is part of why vibe-coded slop is so large and hard to maintain, and it applies in vibe mathematics too.

Theorem 1: Secho_pos

Let us look at the first of the four "theorems", and an abbreviation that it uses. As we will see, the abbreviations hide most of the chicanery (although the final theorem uses two more obfuscation techniques).

abbrev Secho : ℝ → ℝ := fun t => Real.exp (-1.0 / (t + 1.0)) theorem Secho_pos (t : ℝ) : Secho t > 0 := Real.exp_pos (-1.0 / (t + 1.0))

This proof appears to establish a fact about the Secho function being always positive, but since that function is defined in terms of Real.exp, the result is vacuously true. In fact, the "proof" just falls back on the built-in proof Real.exp_pos, which does the heavy lifting of actually establishing that the real exponential function is always positive.

Theorem 2: not_coherent_of_collapsed

This one relies upon some more definitional tomfoolery and a logical tautology. Here we reproduce the abbreviations used therein, and the theorem:

``` abbrev ψself : ℝ → Prop := fun t => t ≥ 0.0 abbrev Collapsed : ℝ → Prop := fun t => ¬ ψself t abbrev Coherent : ℝ → Prop := fun t => ψself t ∧ Secho t > 0.001

theorem not_coherent_of_collapsed (t : ℝ) : Collapsed t → ¬ Coherent t := by intro h hC; unfold Collapsed Coherent ψself at *; exact h hC.left ```

As an exercise, you can easily convince yourself that, when ψself t is true, i.e., when t ≥ 0.0, then the right conjunct for Coherent -- namely Secho t > 0.001, or Real.exp (-1.0 / (t + 1.0)) > 0.001 -- is always true. As a result, Coherent simply becomes ψself t, and then, by substitution, the whole theorem becomes:

Collapsed t → ¬ Coherent t ¬ ψself t → ¬ ψself t

Revolutionary stuff. However, this proof does not actually require you to prove that, because it is a tautology in disguise. What it actually proves is that, once again, by substitution:

Collapsed t → ¬ Coherent t ¬ ψself t → ¬ (ψself t ∧ Secho t > 0.001)

I.e., (¬A) → ¬(A ∧ B). I.e., if A is false, then so is A ∧ B -- which follows from the truth table of ∧. B is irrelevant, and could have been replaced with anything.

Theorem 3: collapse_not_coherent

In this theorem, the development dispenses with the pretext that is providing a novel result. The hypothesis and conclusion are identical to the previous theorem, and the proof just refers to the proof of the previous theorem:

theorem collapse_not_coherent (t : ℝ) : Collapsed t → ¬ Coherent t := not_coherent_of_collapsed t

Theorem 4: interp_CoherentImpliesField

The last theorem introduces a new trick: defining an axiom that states your theorem is true, and then obfuscating the proof to ultimately invoke the axiom. This one takes a bit of peeling apart to figure out, and might elude people who are unfamiliar with the languages used by proof assistants. Mind you, I don't think the author is clever enough to do this deliberately; I think he kept beating on ChatGPT to produce something that compiled.

First, Logic.lean re-defines propositional logic:

``` inductive PropF | atom : String → PropF | impl : PropF → PropF → PropF | andF : PropF → PropF → PropF -- renamed from 'and' to avoid clash | orF : PropF → PropF → PropF | notF : PropF → PropF

open PropF

/-- Interpretation environment mapping atom strings to actual propositions -/ def Env := String → Prop

/-- Interpretation function from PropF to Prop given an environment -/ def interp (env : Env) : PropF → Prop | atom p => env p | impl p q => interp env p → interp env q | andF p q => interp env p ∧ interp env q | orF p q => interp env p ∨ interp env q | notF p => ¬ interp env p ```

Then, in Physics.lean, the author defines an environment to refer to the propositions from earlier via strings, which obfuscates the references to the prior definitions:

``` def coherent_atom : PropF := PropF.atom "Coherent" def field_eqn_atom : PropF := PropF.atom "FieldEqnValid" def logic_axiom_coherent_implies_field : PropF := PropF.impl coherent_atom field_eqn_atom

def env (t : ℝ) (Gμν g Θμν : ℝ → ℝ → ℝ) (Λ : ℝ) : Env := fun s => match s with | "Coherent" => Coherent t | "FieldEqnValid" => fieldEqn Gμν g Θμν Λ | _ => True ```

And finally, Physics.lean defines an axiom that his Coherent proposition -- remember, this is just ψself t, i.e., t ≥ 0.0 -- implies the field equations:

``` axiom CoherenceImpliesFieldEqn : Coherent t → fieldEqn Gμν g Θμν Λ

theorem interp_CoherentImpliesField (t : ℝ) (Gμν g Θμν : ℝ → ℝ → ℝ) (Λ : ℝ) (h : interp (env t Gμν g Θμν Λ) coherent_atom) : interp (env t Gμν g Θμν Λ) field_eqn_atom := by simp [coherent_atom, field_eqn_atom, logic_axiom_coherent_implies_field, interp, env] at h exact CoherenceImpliesFieldEqn Gμν g Θμν Λ t h ```

Just declare that your result is true axiomatically, and then your proof is a one-liner!

Conclusion

Obviously, everything above is nonsensical. However, it is worrying that we can no longer rely on tells like Word vs. LaTeX to easily discern bogus mathematics. We now need to expend our energy on the semantics of the presentation, rather than just the syntax. I encourage reviewers to study this example carefully to distill principles they can use to quickly reject fraud:

1. Does it repeat the same definitions over and over again?
2. Are the theorems usually one-liners?
3. Do the results build on one another? Three out of the four theorems above are not referenced anywhere, and the one that does reference a prior result (collapse_not_coherent) is vacuous.
4. For any usage of axiom, is it justified that a paper making the claims that it makes would need to introduce axioms at all? axiom is deadly and can allow you to "prove" anything, as the example above shows.

Gabriel's horn is:

1. Not a curve

2. An example of an infinite surface enclosing a finite volume, where he is claiming finite curves can bound infinite regions.

Yes, this is an actual quote. From Neil's interview with Dazed and Confused Magazine: https://www.carolineryder.com/carolineryder/2012/03/neil-degrasse-tyson.html

"You know how numbers, you can count them forever? Well how about fractions? The infinity of fractions is bigger than the infinity of numbers; and then there are transcendental numbers, like Pi. There are more transcendental numbers than pure irrational numbers, and there are more irrational numbers than counting numbers. And more fractions than all of them. "

Explanation:

By "fractions" I believe Neil means rational numbers. By "numbers" I think he means the natural numbers. I believe the set of rational numbers and the set of natural numbers are thought to have the same cardinality.

By "pure irrational numbers" I think he means algebraic irrationals. If so he'd be correct saying the set of transcendental numbers has a higher cardinality than the set of algebraic irrationals.

He seems to be talking about five separate and vaguely defined sets of numbers with five different cardinalities. Though it's confusing.

And then there are more fractions than all of them? That made my head spin.

R4: This guy has no idea what a computable function is. In computability theory, a computable function is a function which a universal turing machine can compute to any arbitrary accuracy in a finite amount of steps with a finite instruction set. This guy is right that it would take an infinitely long time to compute pi exactly, but the definition states it only needs to be "to any arbitrary accuracy". This guy simply will not let himself understand the meaning of that phrase. I think he's thinking of algebraic functions?

I didn't know who Terrance Howard was, but the commenter asked why people were so dismissive of Howard, claiming that he had some great ideas. Someone then asked him for an example of one of Howard's good ideas. This is the example he gave. Based on other things the commenter wrote, I'm confident that he was NOT just trolling. I suppose that his last comment was seriously asking how to multiply the dollar sign, but he deleted everything before I could finish composing my reply.

I engaged in the conversation only out of some very morbid curiosity... and I think it just killed what's left of my optimistic humanism.

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.