@bemmesr@mathstodon.xyz

bemmesr @bemmesr@mathstodon.xyz · 6d ago

Replying to on mathstodon.xyz

Hello again @SmartmanApps@dotnet.social Our mutual friend @FishFace@ioc.exchange has discovered an example of a textbook which explicitly states that multiplication must occur before division. Further, this book was published in the sixties, well within your ‘150 year’ timespan: https://archive.org/details/algemene-rekenkunde/page/44/mode/2up?q=volgorde+bewerking I'm curious about this, since I recall you making the claim that there is no such textbook published within the last 150 years. I'm wondering if you consider this as a textbook or maybe you would dispute its meaning? For convenience I have written out the relevant portions to be translated here: ‘Indien verschillende bewerkingen achter elkaar moeten worden uitgevoerd, dient dit te geschieden in onderstaande volgorde: Machtsverheffing, Worteltrekking —logaritmeneming, Vermenigvuldiging, Deling, Optelling — Aftrekking. Een bewerking, die hoger geplaast is gaat vóór een lager geplaatste bewerking. Worteltrekking en logaritmeneming zijn gelijkwaardig, evenals optelling en aftrekking.’

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bemmesr @bemmesr@mathstodon.xyz · Aug 03, 2026

Replying to @SmartmanApps@dotnet.social

@SmartmanApps@dotnet.social I'm afraid you'll have to educate me here, I'm not familiar with the technique. Perhaps there is a resource you can point to which contains a definitive series of proofs for the fundamental laws of arithmetic, which do not rely on any axioms? Or, if not, perhaps you could offer guidance on how I might construct such a proof system?

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bemmesr @bemmesr@mathstodon.xyz · Aug 03, 2026

Replying to @SmartmanApps@dotnet.social

@SmartmanApps@dotnet.social you say that arithmetic is ‘literally proven’, which I do agree with to some extent. For example, the Principia Mathematica by Whitehead and Russell proves much of the foundations of modern mathematics, but it does use some axioms to begin with. That is, it doesn't just prove these things out of nowhere, but builds up each proposition from some basic *assumed* facts. My question then is, are you suggesting that you can prove mathematical truths without any axioms? If so, could you provide an example of such a proof?

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bemmesr @bemmesr@mathstodon.xyz · Aug 02, 2026

Replying to @FishFace@ioc.exchange

@FishFace@ioc.exchange this is where the Socratic method is supposed to shine. Instead of attempting to lay the groundwork yourself, have your interlocutor explicitly state their assumptions, recursively, until an error is revealed. Unfortunately, almost everyone holds inconsistent beliefs, so the technique doesn't necessarily say as much about the subject matter as it does about the nature of human belief. Still, it's tempting to try it. I don't really know how effective it is in practice...

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bemmesr @bemmesr@mathstodon.xyz · Aug 02, 2026

Replying to @FishFace@ioc.exchange

@FishFace@ioc.exchange indeed! It's a good exercise in debate. Just when you've cut off one head another appears elsewhere... Though I suppose a better debater would know how to manage those heads and to force their interlocutor to acknowledge the dead ones, or perhaps would refuse to engage with diversions at all.

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bemmesr @bemmesr@mathstodon.xyz · Aug 02, 2026

Replying to @FishFace@ioc.exchange

@FishFace@ioc.exchange I see, well he should have to answer to that if he really wants to hold on to the 150 year timespan. I may present this to him later if I fail to find such an example by someone other than Lennes.

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bemmesr @bemmesr@mathstodon.xyz · Aug 02, 2026

Replying to @FishFace@ioc.exchange

@FishFace@ioc.exchange yeah something like that, still, I've read the relevant page and it seems to me that though they demonstrate multiplication then division, they don't seem to explicitly indicate that this order must be followed. Further, they always have multiplication in brackets if it directly follows division, which to me implies that they would expect a different result without those brackets (possibly due to left-right processing)

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bemmesr @bemmesr@mathstodon.xyz · Aug 02, 2026

Replying to @FishFace@ioc.exchange

@FishFace@ioc.exchange

SmartmanApps is on the record as saying this textbook is just wrong, so it won't convince him. When I challenged him that he simultaneously thinks "all textbooks agree" and "Lennes' textbook is wrong" I believe this was the point where he said that, since his sentence was in the present tense, he had not meant to include Lennes. I never did work out exactly what timespan of publication he includes under "all textbooks agree." Perhaps it's a vacuous truth, unless at the very moment of utterance, a texbook on the subject is being written. Or printed?

I did notice this, so I held off on using it as a source. Thankfully he has provided ‘150 years’ as a timespan, so I'm still searching for a source within that period. On this search I've encountered the general idea that the division operator should be avoided over and over, but have yet to find definitive examples of textbooks which use a different order of operations, despite that being a cited reason for avoiding the symbol.

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bemmesr @bemmesr@mathstodon.xyz · Aug 02, 2026

Replying to @bemmesr@mathstodon.xyz

@SmartmanApps@dotnet.social I have been looking for the textbooks as you requested. I've found numerous sources which say that there is such a discrepancy, though I have yet to see actual examples of it. Meanwhile, while I continue to search, I'd suggest you read https://math.berkeley.edu/~wu/order5.pdf as it essentially makes the same argument I'm making, perhaps with more clarity and conviction. It's not an uncommon idea. In ‘Mathematics From the Birth of Numbers’ by Jan Gullberg, on page 122 under ‘Operational Precedence’, Gullberg says, ‘we also note the left‐to‐right rule of precedence for multiplication and division: 24/2×6... but as this principle of precedence is not universally respected, we recommend, for the first case: (24/2)×6...’ Now, this has turned out to be somewhat of a research task, so I'll at least give you that there are not *many* textbooks published in the last 150 years which differ on the order of operations, but I won't yet concede that there are *none*, since of course we have no reason to believe this to be the case. I'd also like to add that the reason I say that the order of operations is arbitrary is because arithmetic is essentially just an axiomatic system, and of course the interpretation of a given sentence in such a system depends entirely on your definitions and axioms.

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bemmesr @bemmesr@mathstodon.xyz · Aug 02, 2026

Replying to on dotnet.social

@SmartmanApps@dotnet.social while I look for such an example, I'll leave you with this to ponder: https://en.wikipedia.org/wiki/Order_of_operations#mwyA ‘There is no universal convention for interpreting an expression containing both division denoted by '÷' and multiplication denoted by '×'.’

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bemmesr @bemmesr@mathstodon.xyz · Aug 02, 2026

Replying to @SmartmanApps@dotnet.social

@SmartmanApps@dotnet.social alright ‘smartman’, if there is only one standard for order of operations then why is it that when I put ‘3÷2(6)’ into my #Casio fx-100AU PLUS the answer is ‘0.25’, but when I put the same exact string into my Google Pixel's calculator app the answer is ‘9’? I suppose you'd say my #android is wrong, right? Maybe so, but then you can't deny that the confusion is pervasive. If 72.77% (https://www.sci-tech-today.com/stats/android-phones-statistics/) of mobile phones will tell you the wrong answer in their default app, are we simply to ignore it? Well, again, maybe so. That's the way you would have it, isn't it? If they're wrong they're not worth listening to. But when I write a mathematical expression, I want it to be that no matter who copies it down into whatever calculator, that they all get the same result. To me that would be ideal. And to achieve that, I will use as many brackets as necessary (or I will simply use fraction notation if I can). As for brackets being new, this isn't really an argument. We would expect the language of mathematics to become more standardised, and less ambiguous, over time. In the early days of mathematics we didn't even have standard symbols for any of the arithmetic operators. To begin with, mathematics was the privilege of the educated few, and now days it is a common tool for the common person. With that comes a requirement for intuitive clarity, not knowledge of some standard order of operations (which again, a majority of mobile devices don't even adhere to).

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bemmesr @bemmesr@mathstodon.xyz · Aug 02, 2026

Replying to @SmartmanApps@dotnet.social

@SmartmanApps@dotnet.social surely you understand that the interpretation of a given string of symbols depends entirely on your definitions, and surely you can admit that while we are close, we don't have a global standard for these definitions. Even if there was one global standard for order of operations and definitions, people would still be confused about 1÷ab, as you rightly pointed out with your example. And yes, they would be wrong in such a case, but that doesn't magically make the source of confusion go away. The reason I say that fractions make things clearer is because their scope is entirely unambiguous. To write a fraction you are forced to indicate its scope in its entirety, whereas with ‘÷’ we need to indicate its scope by relying on brackets. It's true that fractions are difficult to write in many contexts — though they *can* be written on a single line without brackets, if we use #LaTeX for example, or if it's manuscript. I don't think it's a coincidence that in the majority of maths papers where division is a relevant operation, we tend to see fractions and not division operators. If we really did want to use the division operator — for whatever reason — I would suggest that we use the form ‘((φ)÷(ψ))’, where the brackets leave no room for confusion (however misplaced the confusion may be). This could be excepted in the case where φ or ψ are atomic expressions, since then there is only one way to interpret the order.

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bemmesr @bemmesr@mathstodon.xyz · Jul 31, 2026

Replying to @bemmesr@mathstodon.xyz

@SmartmanApps@dotnet.social I get that you can't just replace a division symbol with a fraction because they don't work the same way, but for any expression which uses a division symbol there exists an equivalent expression which uses no division symbols and may use fractions.

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bemmesr @bemmesr@mathstodon.xyz · Jul 31, 2026

Replying to @SmartmanApps@dotnet.social

@SmartmanApps@dotnet.social well yeah, I know it has different precedence to a fraction. But I believe division operators are just more ambiguous to a lot of people. Fractions make things much clearer for everyone, and in my eyes they should be used instead.

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bemmesr @bemmesr@mathstodon.xyz · Jul 30, 2026

Replying to @SmartmanApps@dotnet.social

@SmartmanApps@dotnet.social I just think it's generally an error to use the division symbol instead of a fraction.

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bemmesr @bemmesr@mathstodon.xyz · Jul 23, 2026

Replying to @lianna@micro.webgarden.click

Mention of psychosis, mention of psychotic triggers, passing mention of drugs Hover or focus to reveal Sensitive

@lianna@micro.webgarden.click okay, thank you. I think I get the idea. As a side‐note, some of those types of delusion you listed I wouldn't generally class as delusional. For example, nihilism is just a philosophy. But, then again, I suppose there's holding nihilistic views and then there's acting on nihilistic views, which could be rather different things.

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bemmesr @bemmesr@mathstodon.xyz · Jul 23, 2026

Replying to on beige.party

@TwoClownsEating@beige.party what a strange expression

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bemmesr @bemmesr@mathstodon.xyz · Jul 23, 2026

Replying to @lianna@micro.webgarden.click

Mention of psychosis, mention of psychotic triggers, passing mention of drugs Hover or focus to reveal Sensitive

@lianna@micro.webgarden.click is there some resource you know of that I can use to check if any of the topics I'm posting about are likely to be someone's trigger?

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bemmesr @bemmesr@mathstodon.xyz · Jul 22, 2026

Replying to @FishFace@ioc.exchange

@FishFace@ioc.exchange you've explained it well. That D and E could be any number of expressions which cancel out to the overall original numerator when cross‐multiplied is good enough reason to avoid doing it this way. I think I understand where my confusion was coming from now but I need to investigate a few ideas to know. Thanks for your help!

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bemmesr @bemmesr@mathstodon.xyz · Jul 22, 2026

Replying to @FishFace@ioc.exchange

@FishFace@ioc.exchange sort of. I wouldn't say ‘just’ do this, since I'm thinking my approach is actually harder, because we'd be ignoring information we have (as in, we know that the left numerator is a constant and that the right is no more than order one), but out of curiosity, what would happen if we went with this order‐agnostic approach where we simply let the numerators stand in for any polynomial in x? I would expect that we get the same final answer, even if it takes longer. So yeah, let the left numerator be called E(x) and the right D(x), as you say.

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bemmesr @bemmesr@mathstodon.xyz · Jul 22, 2026

Replying to @FishFace@ioc.exchange

@FishFace@ioc.exchange that makes sense, thank you. It's a choice to limit the placeholders to constant values. I still don't quite understand why the technique doesn't work if we don't make this restriction though. After all, there ought to be a way to discover which polynomial A(x) would satisfy the equation between the original polynomial fraction and our decomposed sum of fractions, wherein A(x) is one of the numerators. I could appreciate that in fact the technique is easier if we do make this restriction but I still would like to know where it fails otherwise. Surely we could still achieve those nice constant numerators in the case of linear denominators if we approached it this way, since a constant is also a polynomial, just of order zero.

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bemmesr @bemmesr@mathstodon.xyz · Jul 21, 2026

Replying to @SmartmanApps@dotnet.social

@SmartmanApps@dotnet.social it makes sense that the textbook should have very dry material and should rely on a teacher to elucidate its contents to their students. Though, many teachers have a pretty uninvolved style of teaching, where they hand out the material without much comment. As you suggested, I had a particular problem in mind when I wrote my post. We are told that when splitting a complex polynomial fraction, we are to split the denominator into its factors, and that for each of these factors we can create a new fraction with a placeholder variable as its numerator and the given factor as its denominator, and we are given to believe that the sum of these new fractions can be equated to the original fraction, so long as we find the right values for the numerators. This is all fairly intuitive, but I have some trouble when considering the scenario where one of the resulting denominators is an irreducible quadratic. We are told that in such a case, the numerator should be a prototype of linear form, with two place holder variables in the form Ax + B. I've managed to get my hands on something of an explanation for why this is necessary, which is that we want to express as many polynomials as we can in the numerator, but any polynomial of order two or higher will make the fraction representable as the sum of a polynomial and a fraction with an order one or fewer numerator. Supposedly this means we don't need to bother with more than order one. What I don't understand is why we can't just use a single placeholder variable? After all, a variable could well come to be defined as being a polynomial in some other variable, so why doesn't this work?

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bemmesr @bemmesr@mathstodon.xyz · Jul 20, 2026

Replying to @bemmesr@mathstodon.xyz

@rzeta0@mathstodon.xyz if you were a cat, as you mentioned, you would not be a man, but still a mammal. This doesn't violate ‘I am a man only if I am a mammal’, nor does it contradict ‘if I am a man then I am a mammal’, it's a different scenario.

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bemmesr @bemmesr@mathstodon.xyz · Jul 20, 2026

Replying to @rzeta0@mathstodon.xyz

@rzeta0@mathstodon.xyz if it's really true that whenever proposition X is true, then so is Y (which is what X→Y asserts), then it cannot possibly be that X is true without Y being true. ‘X is true only if Y is true’. It doesn't mean that X will be true if Y is true, just that it can only be that X is true if Y is true. To tie this in with your chosen example, it's obviously true that if I am a man then I am a mammal, and it's also true that I cannot be a man unless I am a mammal. Really, ‘if X then Y’ and ‘X only if Y’ are two ways of saying the same thing. I'm not sure if that makes sense.

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bemmesr @bemmesr@mathstodon.xyz · Jul 20, 2026

I've been taking some time to study some of the #mathematics that I learnt in high school and subsequently forgot, and I'm beginning to remember one of the big reasons I found the whole thing rather frustrating: so many #maths resources explain the techniques you can use to achieve certain results, but never bother to explain why these techniques work. It seems that I need proof that a given technique works before I can really trust it, and until then my brain refuses to accept it into the canon.

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bemmesr @bemmesr@mathstodon.xyz · Jul 10, 2026

Replying to @mid_kid@fosstodon.org

@mid_kid@fosstodon.org I like dmenu

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bemmesr @bemmesr@mathstodon.xyz · Jul 05, 2026

Replying to @DrorBedrack@mastodon.social

@DrorBedrack@mastodon.social I suppose because under other systems the government would provide cheap or free education, so the need for student loans wouldn't exist.

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bemmesr @bemmesr@mathstodon.xyz · Jun 25, 2026

Boosted by

Welcoming committee @welcome@friends.deko.cloud

#Introduction Hello everyone, I finally found a home for my profile outside of mastodon.social, after nearly two years of being on the network! #Mathstodon was the natural choice for me because I've always had a casual interest in #mathematics and lately I've been getting into (basic) formal #logic I worked as a #software #engineer for nearly two years, but after moving countries I found myself without a job for quite a while. During that time, I ended up learning a lot of things I don't think I would have had the energy for had I still been working. On many matters my opinion has changed substantially over the last year or so, and perhaps for the better, but on the whole I've gone from unearned confidence to general uncertainty, which I know some find irritating but I think it leads to interesting discussions. I'm excited to see what this instance has to offer and to hopefully make some meaningful contributions. — Bemmesr

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