Mastodawn

1/9
#MathsMonday #Mathematics
This rubbish article https://www.scientificamerican.com/article/mathematicians-cant-agree-on-whether-0-999-equals-1/ popped up in my feed a few times, and I've already debunked the various points, but will cover it with specific links for each (non-)point.

"Mathematicians can’t agree on whether 0.999... equals 1" - yes they can, it's not, as per division, limits, infinite decimals, and other #Maths topics, all found in #Math textbooks

"by Manon Bischoff" - "is a theoretical Physicist". Maybe just stay in your lane dude... 🙄

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 13

2/9
"Countless debates in classrooms" - there are no debates in classrooms. Students are taught all of the topics which prove they're not equal.

"Teachers, professors and math-savvy Internet users repeatedly affirm that it does" - nope! Teachers most definitely do NOT say they are equal. Dude has clearly not ASKED any teachers, and is outright just making this up!

"many others still refuse to believe them" - people who can show you actual proofs and textbooks... 🙄

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 13

3/9
"the decimal representations of some fractions are infinite, such as 1⁄3" - which we know are APPROXIMATIONS https://dotnet.social/@SmartmanApps/116316765092652161

"one therefore chooses a symbol because a decimal notation would only approximate the actual value" - that's true of ALL infinite decimals. You so nearly had it!

"1⁄3 corresponds to the decimal number 0.333...." - corresponds, but is only an approximation...

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱 (@[email protected])

Attached: 1 image 1/5 #MathsMonday #Mathematics #Math This week I'm coming back to the topic of 0.(3) only being approximately equal to 1/3, which I discussed previously at https://dotnet.social/@SmartmanApps/115207065189190900 At the time, I knew this was true simply from knowing doing the division always leaves a remainder of 1. Since then I've now seen 2 #Maths textbooks which explicitly spell this out, that all non-terminating decimals are only approximations, and that only terminating decimals are exactly equal to fractions...

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 13

4/9
"multiply it by 3 to get 0.999...." - which goes against the rules of Maths, hence using symbols for them instead, such as pi https://dotnet.social/@SmartmanApps/115603725411319763

"They reason that because 1⁄3×3=1" - and that reason is wrong. See above

"there are a few other proofs" - that WASN'T a proof. You violated the rules of Maths, meaning it's not a proof https://dotnet.social/@SmartmanApps/115603723212470958

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱 (@[email protected])

3/6 Let x=0.(9) (multiply both sides by 10) 10x=9.(9) (substitute 0.(9)=x) 10x=9+x (subtract x from both sides) 9x=9 (divide both sides by 9) x=1 1. in the first place, we can't do arithmetic with infinitely recurring numbers. This is why, as I discussed at https://dotnet.social/@SmartmanApps/115365301522477363, the sum of the infinite series is DEFINED as being the limit of the series, because the limit is finite, and thus we can do arithmetic with it, but it's only an approximation of the infinite sum...

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 13

5/9
"you have what’s called a geometric series, something mathematicians have known how to solve for several hundred years" - what we ACTUALLY solve is the LIMIT of the series, given the infinite sum can't actually be calculated https://dotnet.social/@SmartmanApps/115365301522477363

"the term 1⁄10n becomes zero" - no it doesn't. It can literally NEVER be zero - that's exactly what makes it a limit! See previous link. 0/∞=0, 1/∞ doesn't. A non-zero numerator means we have a non-zero fraction. This is so not-complicated!...

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱 (@[email protected])

Attached: 1 image 1/5 #MathsMonday #Mathematics #Maths #Math So we've seen that 0.(3) is merely a decimal APPROXIMATION of 1/3, due to the limitations of using Base 10. It's also true, for the same reason, that 1 is merely an integer APPROXIMATION of 0.(9), since 9 is infinitely recurring. If we think of 0.(9) as being 0.9+0.09+0.009+... this leads some to make a false equivalence argument, which is wrong in 2 ways...

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 13

6/9
"This example is just one of many proofs" - that ALSO wasn't a proof, for the reasons I just gave.

"if we switch to binary notation, which consists only of 0’s and 1’s, the same problem arises" - the problem being your lack of familiarity with Maths 🙄

"Even though mathematics is a subject in which you can derive correlations exactly, with minimal room for interpretation" - and yet here you are making up your own interpretations... 🙄

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 13

7/9
"if you look at the number line and pick any two numbers, there are always infinitely many more between them" - no there isn't https://dotnet.social/@SmartmanApps/115682175610484093

"You have found a break in the number line" - no you haven't. Numbers are discrete, there are no gaps between them. See previous link

"As soon as you calculate a sum, you have to round up" - no you don't. Calculators do https://dotnet.social/@SmartmanApps/115207066209800987, but there's no such rule of Maths. Again you're making things up...

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱 (@[email protected])

Attached: 1 image 1/7 #Maths #Math This #MathsMonday Why there isn't an #infinite amount of #numbers between 0 and 1 on the #numberline AKA The number-line isn't a TARDIS! Firstly, one thing to point out is that people often conflate numbers with #numerals. This is important because #scalars also use numerals. Even though both numbers and scalars use numerals, they aren't the same thing! In #Mathematics, especially #arithmetic, we use numbers to count things. They enumerate how many things there are...

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 13

8/9
"This rounding up also applies to multiplication, such that 0.999...×1=1, which means a basic rule of mathematics, that anything multiplied by 1 is itself, no longer applies" - it's an Identity in Maths, that ALWAYS applies. 0.(9)x1=0.(9). 🙄 This dude sure does like making up imaginary rules of Maths - here the dude uses an imaginary "rule" of Maths to claim that an actual rule of Maths doesn't apply!...

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 13

9/9
"nonstandard analysis, which allows for so-called infinitesimals" - so does high school Maths, used in all the actual proofs that they aren't equal

"if they differ by one infinitesimal" - they do! 😂

"no more so than conventional calculus" - which ALSO says they aren't equal

"there is still a debate whether 0.999... = 1" - no there isn't. It's a fact of Maths that they aren't equal

"working with the numbers and calculation familiar to most of us" - but apparently not familiar to you 🙄

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Hugh Simpson Apr 13

@SmartmanApps a field that includes real numbers, but excludes infinitesimals, is entirely consistent...? It's not like this is some hard-hitting truth to mathematics? Weird to me you accept infinitesimals but previously have denied the arithmetic of infinite series, essentially claiming that S(0,inf,9*10^(-n)) - S(1,inf,9*10^(-n)) should be illegal. I'm not sure how you can retain interesting or useful properties in such a system. Anyway 0.(9) is clearly the sum of the latter series. Which is ?

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 13

@silasmariner
"real numbers" - Reals aren't numbers, they're scalars https://dotnet.social/@SmartmanApps/115840140785226744

"have denied the arithmetic of infinite series" - no I haven't. We calculate the limit

"S(0,inf,9*10^(-n)) - S(1,inf,9*10^(-n)) should be illegal" - not should be, is, that's why we calculate the limit in it's place for arithmetic

"Anyway 0.(9) is clearly the sum of the latter series. Which is" - not able to be calculated, so we calculate the limit instead. Not sure how many times I need to say that

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱 (@[email protected])

Attached: 1 image 1/5 This #MathsMonday I'm expanding on #scalars in #Mathematics. Previously we discussed that integers in #Maths are discrete, and used for counting things. In #Math we also have the Real Numbers, which is a bit of a misnomer since we don't usually use them for counting! It's more a case of people conflating them with numbers, because both use numerals. Rather, the Reals are used as multipliers, to provide a scale (though sometimes used as a fraction's decimal equivalent)…

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Hugh Simpson Apr 14

@SmartmanApps your usage of 'scalar' vs 'number' is not one I've ever seen before, I wonder where you picked it up from. What is the distinction, to you? For me I have only seen the use of scalar vs domain element in the realms of linear algebra (scalar vs vector) -- never before have I heard the claim that a scalar is something different to a number...

Anyway we don't calculate the limit in its place -- the limit of a sum of an infinite series is its value, proof by reductio ad absurdum.

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Hugh Simpson Apr 14

@SmartmanApps not that you need it in the instance of that particular sum, because the elements literally cancel out; that's what makes it so easy to determine the limits of the original sum for such series. This is how people do maths, it is _ubiquitous_.

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Hugh Simpson Apr 14

@SmartmanApps are you, perhaps, of the opinion that there should be no such thing as infinite collections in mathematics? That one cannot talk freely of, say, the set of all integers? Would you dispute the Cantor diagonalisation argument and proclaim that there is the same cardinality of reals as rationals because anything over finite is just unmanageable? That transfinite induction is an illegal technique? Genuinely curious.

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Hugh Simpson Apr 14

@SmartmanApps ah I've followed your link now. I see that you do indeed deny the existence of the infinite. Finite constructivism or something I think that school's called? It's pretty niche. Odd that you're so adamant on what you must know is a minority position, but nonetheless fascinating. Alas, most of my favourite mathematics (complex analysis! Set theory!) all relies rather heavily on the manipulation of the infinite, so I suspect we'll find little to agree on.

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Hugh Simpson Apr 14

@SmartmanApps I will add, finally, that I do however harbour a soft spot for model theory, so your model is interesting. Philosophically, however, I think it's wrongheaded to insist on some models being Platonic truths and others being nonsensical -- internal consistency is sufficient IMO, albeit obviously unprovable within a system (_thanks_ Gödel, you ruined a perfectly lovely dream). We can apply the maths we dream of with infinity in eg electrical engineering so clearly it has merit.

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 14

@silasmariner
"your usage of 'scalar' vs 'number' is not one I've ever seen before" - we teach it in high school 🙄

"I wonder where you picked it up from" - school, textbooks

"we don't calculate the limit in its place" - yes we do

"the limit of a sum of an infinite series is its" - limit

"not that you need it in the instance of that particular sum" - you need it for every infinite sum

"so easy to determine the limits of the original sum for such series" - which is exactly why we do it

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱

@silasmariner
"This is how people do maths, it is _ubiquitous_" - that's right. Calculating the limit is ubiquitous. You haven't even come up with any way that one could actually calculate the infinite sum

"the set of all integers? "- we don't. It's just the Integers. Sets are finite, by definition.

"Would you dispute the Cantor diagonalisation argument" - it's easily proven wrong. That's why Mathematicians said he was a madman 🙄 https://dotnet.social/@SmartmanApps/116276431640577565

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱 (@[email protected])

Attached: 1 image 1/6 Further extending this #MathsMonday on the laughable Cantor #Mathematics claim that the "infinite sets" of the Naturals and the Evens are "the same size", let's accept that #Math argument for a moment, and prove by contradiction that it can't be true... If sets can be infinite, that means not only do they come with the #Maths bracket form of set notation, and cardinality, but also set operations, like Union and Intersection, and ways to depict those sets/operations... with Venn diagrams...

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 14

@silasmariner
"I see that you do indeed deny the existence of the infinite" - says person failing to point out anywhere where I did any such thing. 🙄 We use it in calculating limits for starters

"Odd that you're so adamant on what you must know is a minority position" - odd that you call a majority position "minority". 🙄

"internal consistency is sufficient IMO" - and Cantor's "proof" is inconsistent with the rules of Maths, hence why Mathematicians called him a madman 🙄

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Hugh Simpson Apr 14

@SmartmanApps dude the axiom of choice in ZFC - the 'standard model', if you will - necessitates understanding of and accord with the uncountability of the reals. Cantor is not like some weird kook, he's foundational

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 14

@silasmariner
"uncountability of the reals" - because they're continuous, not discrete like numbers, as taught in high school, as I already said. 😂 This was all well-known well before Cantor came along to try and abuse it https://dotnet.social/@SmartmanApps/115840140785226744

"he's foundational" - yet somehow manages to never get mentioned at any time in school, where we teach the foundations of Maths, including the rules used with sets 😂

"not like some weird kook" - Mathematicians said otherwise

💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱 (@[email protected])

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Hugh Simpson Apr 14

@SmartmanApps you absolutely do not teach the foundations of mathematics in schools. You teach the foundations of mathematical intuition, you teach people to count and multiply. You do not (evidently) teach axiomatic frameworks, the notion of well-foundedness, or the actual methods used to justify our inferences re e.g. limits. There is inherently something intuitional about maths, and sometimes your intuition is wrong. Maths is fundamentally logical implication, not epistemic statements...

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Hugh Simpson Apr 14

@SmartmanApps the entirety of the real number line can be defined as isomorphic to set-theoretic constructions, provided one accepts infinite sets, but we sure as hell aren't teaching that in most schools

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Hugh Simpson Apr 14

@SmartmanApps the thing that galls me the most is that standard mathematics - the sort that you can actually do stuff with - hugely simplifies the cognitive leap by some incredibly elegant abbreviation of tedious thoughts that let us just say 0.(9)=1 (for example) so as not to leave etas and deltas lying around superfluously, and you characterise your rejection of this as obvious and simple and accepted and it is anything but. You are hugely overcomplicating the whole endeavour to your detriment

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 14

@silasmariner
"you absolutely do not teach the foundations of mathematics in schools" - that's hilarious dude. 🤣🤣🤣 Arithmetic, algebra,, calculus, trig, sets,...

"you teach people to count" - which is used in things like... finding the cardinality of a set

"e.g. limits" - yep, we teach limits too, and asymptotes, hyperbolas,...

"the entirety of the real number line" - reals don't belong on the number line - it's right there in the name that it's for numbers!

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 14

@silasmariner
"we sure as hell aren't teaching that in most schools" - because it violates the rules of Maths, which we DO teach in schools 🙄

"you characterise your rejection of this as obvious and simple and accepted" - as seen in Maths textbooks

"You are hugely overcomplicating the whole endeavour" - the difference between an infinite sum and the limit of it is quite simple actually - we see it in graphs of a hyperbola for example

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 14

@silasmariner
"you refuse to accept that (0.9 -0.9) + (0.09 -0.09)... = 0" - what you mean is YOU refuse to accept that the LIMIT is the number that the series literally NEVER REACHES 🙄 https://dotnet.social/@SmartmanApps/115365301522477363

"what is it about that that troubles you" - violating the rules of Maths, ignoring literal proofs, need I go on? 🙄

If you don't want to accept what's in Maths textbooks about limits I'll just end up blocking you dude - you've wasted enough of my time already - so ball's in your court...

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Hugh Simpson Apr 15

@SmartmanApps ... No finite truncation of an infinite positive series reaches the limit, true, is that what motivates you? Then you must believe Xeno's paradox to be unresolved. If I halve the distance between Achilles and the tortoise every time, I have an infinite sum. Nonetheless I believe Achilles reaches the tortoise. You, however, must deny this, and consequently believe motion in space to be impossible. It is an absurdity to permit the infinite but deny the implications. Most strange.

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Hugh Simpson Apr 15

@SmartmanApps anyway you need to stop appealing to 'the laws of mathematics' without referencing anything specific, because it just makes you look like you don't understand anything you're talking about. Adding infinite series together is clearly permitted by associativity and commutativity. These are axiomatic, and can only be denied in the context of sums by forbidding the infinite to exist; that you do one but not the other leaves you in an incoherent state

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 15

@silasmariner
"No finite truncation of an infinite positive series reaches the limit, true" - let me fix that for you, no infinite series ever reaches the limit, full-stop

"you must believe Xeno's paradox to be unresolved" - no, it's not a paradox to begin with, since you can't slow time 🙄

"I have an infinite sum" - no, you have an impossible situation where you have slowed time

"believe motion in space to be impossible" - no, slowing time is impossible 🙄

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 15

@silasmariner
"you need to stop appealing to 'the laws of mathematics' without referencing anything specific" - I have specifically referred to limits/asymptotes, not allowed to do arithmetic with infinite decimals, cardinality of sets, etc., and you are pretending I haven't, like the Gaslighter that you are

"it just makes you look like you don't understand anything you're talking about." - says person who has repeatedly shown he doesn't know what he's talking about, such as what limits are

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱Apr 15

@silasmariner
"Adding infinite series together is clearly permitted by associativity and commutativity" - adding their LIMITS together is permitted. Again, there's no way to calculate the actual infinite sum, hence why we use their limits, but you refuse to acknowledge the difference between the infinite sum itself and the limit, despite me posting textbooks that explicitly spell out the difference, so as promised, time for you to go now Gaslighter. Bye now

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Fish Face Apr 14

@silasmariner I'm still dying to know which "maths textbook" says you can't subtract 0.(9) from 9.(9) and get 9. Because our school textbook certainly taught that you could. BBC bitesize (easier to find online, written by teachers) says you can.

What do you think he thinks is the limit of the sequence (0.(9), 0.(9), 0.(9), ...)??

Sadly he hasn't given any new information about where he sits on the finitism front. "No set of integers, but 'the integers' exist" 🤷

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Hugh Simpson Apr 15

@FishFace I think I've run out of patience with this dude, lol. The main thing seems to be some weird misunderstanding around limits, and just absolutely hating Cantor and trying to characterise him as some madman who continues to be rejected by the maths community (as opposed to, like, an absolute lad). 'No one shall expel us from the Paradise that Cantor has created', Hilbert said of him (I looked the quote up, couldn't quite remember it) but our mad m8 would if he could.

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Fish Face Apr 15

@silasmariner oh there's more than that he gets wrong - it's a fascinating world. He also thinks that the order of operations can be proved, and a month ago he was arguing with someone that f(n) = 2n doesn't define a function.

But I don't blame you for losing patience.

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Hugh Simpson Apr 15

@FishFace oooooh that's interesting, I assumed he'd _love_ functions. Usually the kookier mathematicians are heavily into programming and can't operate outside of its restrictions, so they lean heavily on functions... Anyway I appear to have baited enough that a block is imminent. Woe is me.

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Fish Face Apr 15

@silasmariner yeah it's odd. And he is into programming! Apparently not enough to understand all of a function at once. I think it's because he can't go beyond the diagrammatic representation of functions in high school where you exhaustively list ALL pairings.

Enjoy your block, you'll be joining a ... Non-exclusive club, lol. Though sometimes he threatens it a while before following through ;)

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Fish Face Apr 15

@silasmariner oh oh, his latest post has information I haven't seen before!

It's infuriating when he's asked for references and just repeats terms that he uses wrong (e.g. he always says a sequence never reaches its limit but this is false for many sequences; he refers to recurring decimals as "hyperbolas" but even passing to the sequence, it's 1/x^n not 1/n) instead of giving one containing these so called "rules of maths".

But I am really intrigued whether the textbook (I think that's Chrystal) is coherent on sequences and limits. What does it, and he, think the "last term" is? A sequence has order type omega, so there's no last term.

A surprising amount of what he gets wrong does have basis in textbooks from the turn of the 19th century, but it's truly bizarre he doesn't understand what's in textbooks published later. We'd never refer to infinitesimals in textbooks today, nor to the supposed "last term". Unfortunately he doesn't understand the difference between this and standard analysis, and the books don't develop it precisely enough to know which of the different non-standard analyses it could be.

In any case, looks like you're out of time. (And a gaslighter - anyone who disagrees with him persistently is one, it seems). Hope it was fun!

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Hugh Simpson Apr 15

@FishFace it was utterly infuriating, especially the ramble about slowing time in response to Zeno's paradox (which apparently I misremembered the spelling for). I had a feeling that pointing out that particular issue would cause a crisis because it's quite the gotcha, and I had such a good comeback to that. Alas, it will be lost in time. Like tears, in rain. Still, probably all for the best. Initially it was interesting enough, but the condescending tone put me in the mind of a smug teenager

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Hugh Simpson Apr 15

@FishFace I'll be interested to see what you find if you do end up digging out that textbook. Glad if I was able to help in your thankless task of unpicking this timecube-esque idiosyncratic world view

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Fish Face Apr 15

@silasmariner yeah he has a way of going off in weird directions. But even if you catch him out good and proper, he'll just deflect, deny, distract. When I first engaged him, I got to the point where I set out explicitly how he could show good faith by explicitly citing one of several different things. He failed to, but insisted that what he had replied with was such an example. He's so far gone that he can't admit that a page in a book is or is not explicit about something.

If he'd blocked me at that point I probably would never have got to the point of munching popcorn whenever anyone replies to him. I have to wonder if the constant drip of disagreement even registers as odd to him.

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Hugh Simpson Apr 15

@FishFace probably thinks we're all failing to understand limits because of course a finite subset of terms doesn't reach the limit unless the series is 0 after a point. Wonder what he'd make of alternating series. Like, is 1/2 - 1/4 + 1/8... above or below 1/3? Damn it. If only I'd had more time!

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Fish Face Apr 15

@silasmariner found it https://archive.org/details/advancedalgebra01collgoog/page/n208/mode/2up but this is before the section in the same textbook which deals with limits. So this argument cannot be understood formally.

But a commonality among these old textbooks is that they *do* treat recurring decimals as a series rather than the number and thus say "find the sum of 0.999...” or whatever.

Others wholly contradict what he says about the limit not being the sum; if he examined his books honestly he'd realise these are different ways of formalising what's going on, and that formalising 0.999... as 1 is one valid way.

On his recent thread he actually engages with this (there's a textbook screenshot which talks about how we "say that [the sum] is equal to [the limit]", arguing that this language means it's not REALLY true.

What I don't get is how he then says that the infinite sum is something, but different from the limit. I can understand "it's undefined, because addition is only defined for finitely many addends" but that's not the same thing.

Advanced algebra : Collins, Joseph Victor, 1858-1943 : Free Download, Borrow, and Streaming : Internet Archive

Book digitized by Google from the library of the University of California and uploaded to the Internet Archive by user tpb.

Internet Archive

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Hugh Simpson Apr 15

@FishFace what's weird is that that textbook explicitly asks you to sum many infinite series as worked examples. The language clearly implies that this is a permitted operation. So he's cherry picking even within that source. The exegesis is going about as well as I expected lol

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Fish Face Apr 15

@silasmariner I'm continually surprised by how he'll latch on to one thing ("the last term" here), and then ignore *everything* else to the point of inconsistency. "The last term" of an infinite series only really makes sense as an informal argument after already having used it for a finite series, but he takes it as gospel. When a modern textbook says 1/3 = 0.333..., *that* can't be literally true but must instead be informal, because it contradicts the gospel. In spite of the fact that the "last term" thing can't be formalised without loads of extra maths except as a limit! (By that I mean, you could define infinite sequences with a last term as functions on omega+1, but to do so with a single rule you need to define arithmetic involving omega, which he of course hasn't done - all the books agree about ∞ [the only symbol he has to stand for omega] not being a number and a/∞ not being a representation of literal division)

In spite of all that, I find it a very fun challenge to work out where exactly he goes wrong. I go back and forth on whether he's just using mathematical language wrong but underneath believes something consistent, whether his understanding of the real numbers is standard, nonstandard or incomplete, etc.

He says infinitesimals are used in school (I mean, not any more) but does he think they have the same status as like, pi? Does he think 1 - 0.999... is defined? Does he think (1 - 0.999...) / 10 < 1 - 0.999...? It's not clear whether these are forbidden by "rules of maths" that he's never cited, or whether they yield numbers like "0.(0)1".

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Hugh Simpson Apr 14

@SmartmanApps As far as denying infinity... well, like, you refuse to accept that (0.9 -0.9) + (0.09 -0.09)... = 0. So if not infinity, what is it about that that troubles you?