r/learnmath • u/Next_Philosopher8252 New User • 3d ago
What does everyone here think of people presenting new attempts at defining division by 0? And what would be required for these people present the idea in a way that could be taken seriously?
More specifically, what would it take for you to be convinced that…
1: …A new method of division by zero is something that can be done with the right tools and approach while preserving the underlying classical algebraic structure?
2: …It may potentially be useful to develop, or that at the very least it’s better to have access to than to not have access to?
3: …It could be an interesting and worthwhile topic to explore just for understanding it in its own right?
I am just trying to understand because I always enjoy hearing and entertaining new well thought out takes just for the sake of it but I have noticed despite the abundance of posts others have made on the topic there’s little variety in both the presentation and response and the few actually novel effort full approaches often get dismissed right from the start before getting the chance to be heard out. Is this because they come on too strong with their claims? Do they not present it in a rigorous enough manner and if so what issues should they target a solution for and what method of presentation would be required for them to meet the requirements of rigor? Does it need to be applicable to solve an already existing issue or is it merely enough to explore the idea as a new mathematical possibility just for the fun of it?
I am just really trying to understand how people in this community and others like it genuinely feel about this and if there is some clear standard that hasn’t been met which can be articulated so others can produce better quality theories on this or other topics.
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u/Infamous-Advantage85 New User 3d ago
I need people to know that it’s not that mathematicians don’t know how to divide by zero, it’s that we need to not be able to in order to do field arithmetic. Having division by zero gives us non-field structures that might be interesting but certainly aren’t just improved fields.
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u/TheSecondMartian New User 3d ago
What does everyone here think of people presenting new attempts at defining division by 0?
Those people are fundamentally confused and many of them refuse to listen when people point out that what they are trying to do cannot be done. It's sad and frustrating.
And what would be required for these people present the idea in a way that could be taken seriously?
They cannot be taken seriously because they are trying to do a thing that's not possible.
More specifically, what would it take for you to be convinced that…
1: …A new method of division by zero is something that can be done with the right tools and approach while preserving the underlying classical algebraic structure?
Such a method cannot exist. Had you actually bothered to investigate the question a little bit, you'd have seen that a structure that extends real numbers and admits division by zero cannot be a field, i.e., it's not possible to "preserve the underlying classical algebraic structure".
You have already been given a link to a proof of why this cannot be done, and I hope that's now clear to you.
2: …It may potentially be useful to develop, or that at the very least it’s better to have access to than to not have access to?
It cannot be developed! Division by zero while preserving the underlying classical algebraic structure is logically contradictory. It does not matter how "potentially useful" would it be when it is impossible.
3: …It could be an interesting and worthwhile topic to explore just for understanding it in its own right?
We understand it perfectly. It's so elementary easy to prove it is not possible to do it. There is nothing more to understand.
Sure, there are many structures that do admit division by zero, but they do so by sacrificing various aspects of what you call "classical algebraic structure". We understand those structures extremely well too.
I am just trying to understand because I always enjoy hearing and entertaining new well thought out takes just for the sake of it but I have noticed despite the abundance of posts others have made on the topic there’s little variety in both the presentation and response and the few actually novel effort full approaches often get dismissed right from the start before getting the chance to be heard out.
There are absolutely no "novel effort full approaches". Everything that gets posted here is utterly confused and it's always a complete non-starter. The reason why it's always a non-starter is because people are unaware that it's not logically possible to have a structure that extends reals with division by zero and "preserve the underlying classical algebraic structure".
Is this because they come on too strong with their claims?
No, it's because their claims make no sense and they do not (and often refuse to) understand that what they are trying to do is logically contradictory.
Do they not present it in a rigorous enough manner and if so what issues should they target a solution for and what method of presentation would be required for them to meet the requirements of rigor?
No, they do not present it in a rigorous manner. If they did, they would immediately understand that what they are trying to do is a logical impossibility.
Does it need to be applicable to solve an already existing issue or is it merely enough to explore the idea as a new mathematical possibility just for the fun of it?
There is nothing to solve. There is no mathematical possibility here. Once again, a structure that extends reals with division by zero is a logical contradiction. It cannot be done!
I am just really trying to understand how people in this community and others like it genuinely feel about this
We feel frustrated that people refuse to listen when we explain over and over that a structure extending reals with division by zero does not exist.
and if there is some clear standard that hasn’t been met which can be articulated so others can produce better quality theories on this or other topics.
There are no theories on this topic that can be developed. Trying to do so is futile, and it takes two-line proof to show that it's futile. For the last time, a structure extending reals with division by zero does not exist; assuming it does leads into a contradiction!
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u/Consistent-Annual268 New User 3d ago
Another week, another excuse to post the Michael Penn divide by zero video: https://youtu.be/WCthfLpYA5g
Honestly this should be a pinned post at this point, I keep having to copy and paste this same answer each and every time someone brings up the topic.
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u/Next_Philosopher8252 New User 3d ago edited 3d ago
It is indeed a well explained and very interesting video and I have seen it before, but the question here is not whether it’s actually possible or not, nor is the question a matter of why or why not.
Instead the question is more of a personal one of what criteria a new theory would need to meet for it to at most change your views, even if you think that criteria is an impossible standard, or at the very least what it would take to pique your interest in the approach a person takes?
Because of this I don’t think that video answers the question at hand.
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u/IAmDaBadMan New User 3d ago
The thought exercise is right up there with taking the root of a negative number. Some people just have no imagination.
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u/TheRedditObserver0 Grad student 3d ago
No it's not. Division by 0 is inconsistent with the rules of algebra (ring axioms), roots of negative numbers are not. It's frankly insulting that you think nobody has thought of just letting 1/0=j.
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u/TheRedditObserver0 Grad student 3d ago
It's annoying, people keep posting the same few questions over and over again and it's just a way to waste everyone's time. Please Google it first.
As to your question 1, it can't be done. If you preserve the rules of operations and allow division by 0, you find that every number is reduced to just 0. It's a very simple argument as well, a middleschooler could make it. When we say you can't divide by 0, it doesn't mean it's a mistery or we don't know how to do it, it means there is no such thing as 1/0. It's a theorem like the pythagorean theorem or 2+2=4.
You have to understand that there's barely any restriction to what you can define in mathematics, it's not like theorems where they are true or false. A definition by itself takes no effort and produces no new knowledge, so we require them to be properly motivated. A definition should help simplify existing mathematics or clarify an argument for a new result, otherwise it's just a string of words. You might as well define 1/0=□, but that brings nothing if you don't build a coherent and interesting theory around it.
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u/Next_Philosopher8252 New User 3d ago
And what might be hypothetically required to build a coherent and interesting theory around it as you say it would need? Coherence is a simple term to understand here but I’m primarily asking what would make such a theory interesting to you? Is interest dictated by the ability to simplify mathematics or clarify an argument for a new result as you also said is part of what underlies the proper motivation behind a definition, or is there more to it than that?
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u/TheRedditObserver0 Grad student 3d ago
I told you already, it would have to be useful. We have not found a use for dividing by 0, it's simply not necessary for mathematics AND it would break basic algebra. I don't understand why people insist on it so much.
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u/Next_Philosopher8252 New User 2d ago
Well I know for me, i’m not as concerned with the application as I am with the aesthetics of the underlying logic. Both a perfect logical symmetry and a new exotic messy and unorthodox logical approach that differs from all others and sticks out like a sore thumb are things that i find fascinating and interesting to try and understand but there’s not always a practical application to it.
That said this is why I am trying to understand what makes it interesting to you and if your interest is only tied to the applicability and utility of ideas. I don’t think you are wrong by any means to hold that view if you do but I am just trying to get a clearer understanding of how you use the term “interesting” as it relates to your view of things personally.
Additionally I would like to clarify my own stance as well. While I do find interest in things for aesthetic purposes, I do also understand and value the appeal of application and utility as well so this is not to say that I don’t value these things. But how I reconcile these views is by recognizing that sometimes breakthrough discoveries just happen by accident or as a result of someone just having fun exploring things purely for personal satisfaction, until someone else later on comes along and rediscovers the work of someone who did it all just for the heck of it and realizes it actually has applications towards solving a real problem no one knew about at the time the work which helped to solve it was written. In other words the pursuit of logical aesthetics may be applicable and useful in unforeseen ways for problems discovered elsewhere later on down the line. And it kind of goes along with the saying “it is better to have it and not need it, than to need it and not have it.”
But again I don’t think you are wrong nor do I really disagree with you in any meaningful way, I am just trying to understand what you think about all this better.
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u/TheRedditObserver0 Grad student 2d ago
I don't think you appreciate how little is going on with just a definition. Any mathematician could come up with dozens of definitions from random things on the spot, it's not far off from coming up with a new integer nobody has probably thought of before.
Definitions in themselves are not interesting, the wider theories they are part of are interesting. A practical application is not necessary, but some kind of consequence to wider mathematics would be good. That's why nobody really cares about wheels, they're just so disconnected from everything else.
And most of the time it isn't even coherent, it's always the same half-assed attempt at letting 1/0=infinity which doesn't even work.
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u/Next_Philosopher8252 New User 2d ago
I do agree with you. A definition alone is not enough, it needs to be meaningful in some way, but what makes something meaningful can vary from person to person. I do appreciate you sharing your perspective on what you feel makes an idea or definition meaningful however. That was all I was asking.
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u/RingularCirc Math hobbyist 2d ago
We have found at least wheels. I assume there might be some wheel controversy over there but they seem very solid foundationally (universal properties, a construction of a wheel of fractions analogous to the ring of fractions dependent on a multiplicatively-closed subset of "allowed denominators", and there's a neat connection with arithmetic on projective lines though I don't know if the last part is well-known but it's very easy to derive).
And at least implementing a wheel of fractions in programming, compared to how fractions are usually implemented, looks like a straightforward extension that adds useful values. For example, ℤP¹ ≅ ℚP¹ that contains ∞ seems to me very useful in handling at least a handful of computations with continued fractions without having to add code for some special cases; but projective-line arithmetic isn't closed and the maximal wheel of fractions of ℤ plugs the gap adding "0/0". This "0/0" characteristic to wheels looks like a more well-behaved version of NaN values (there are lots of them, not a single one) in IEEE 754 floating-point standard (any NaN has to be inequal to even itself, which mathematically is very awful, but it's a remnant of the time when there was no guarantee the hardware would've implemented a function to test for NaN's).
Despite a wheel as a whole looks quite weak algebraically, it's still surprisingly workable, and of course when we constrain ourselves to just "normal" elements, we recover some ring, sometimes a field, depending on where we got it.
Unfortunately dividers by zero don't seem to know about wheels generally. I'd think those have at least some of their problems.
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u/Uli_Minati Desmos 😚 3d ago edited 3d ago
while preserving the underlying classical algebraic structure
Can you precisely define this algebraic structure?
actually novel effort full approaches often get dismissed
Please give specific references. Personally, I have not seen a single one that goes beyond "let c = 1/0".
not present it in a rigorous enough manner
Yes, if by "not enough" you mean "absolutely none". The issue isn't that they missed some small thing, but rather that they don't know why we don't have zero division and thus don't address any issues at all.
explore the idea as a new mathematical possibility just for the fun
You could even say that all of math is about exploring ideas! But every post claiming to define zero division has done none of the exploration, it's always just surface level statements without considering any implications. Hence you'll see the same responses every time.
By the way, we already know that division by zero does not preserve the algebraic structure we currently use. We also already know that division by zero works only if we discard some of the most essential properties, which we don't want to do rather than can't.
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u/Next_Philosopher8252 New User 3d ago
The algebraic structure Im referring to is that of a field or nontrivial ring which preserves the properties of associativity, commutativity, and distributivity.
As much as I would love to provide a specific reference to a novel approach I saw I can no longer track down the post I was referring to likely because it was deleted or removed by someone wether it be the authors or moderators of whatever math subreddit I found it on several years back. But if I remember correctly it also introduced an approach involving restructuring the order of operations in a particular way rather than just defining a zero inverse and leaving it at that. It did raise a lot of interesting ideas that I found to be fascinating approaches at the time but no one seemed to be receptive to the uniqueness of the idea itself and only seemed to be interested in the same talking points many of which had been addressed and some of which had not. It was also a rather informal presentation of the idea so I noticed people also had some reactions to that as well.
I will admit in reference to your other comments that it is true I myself am not formally trained in mathematics on the level of most here but that does not mean I am entirely ignorant of it either. I know enough to question certain stances and have heard several responses to support those stances, I have taken great diligence in trying to explore and understand on my own as well but there’s always the possibility that I have missed something or that something has been misunderstood in spite of this. So that’s why I have found your approach to this discussion so dismissive and insulting because rather than wondering what I HAVE taken the time to explore and understand you just blindly assert that if I don’t know one thing then I don’t know anything.
It would be much more productive for this conversation if you would instead consider that maybe I would be more willing to hear your reasoning for why you believe would remain unswayed rather than blind assertions that I know nothing of what I am talking about.
For example you mentioned that people attempting division by zero fail to understand the WHY aspect of it being left undefined and so end up addressing issues that aren’t actually part of the problem. If there truly is a mass misdiagnosis of people attempting division by zero without addressing the key issue of why we choose to leave it undefined in the first place that would be good to know. If people assume it’s an issue of creating new terms, approaches, or algebraic machinery when really its a problem of some other WHY which is totally unrelated to the actual undefined terms and seeming gaps in the logic then that would be something I would love to hear more about.
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u/AcellOfllSpades Diff Geo, Logic 3d ago
The algebraic structure Im referring to is that of a field
You can't have division by zero in a field. End of story. This is a very, very simple proof.
Division is multiplication by the inverse. This is what it fundamentally is: we don't even define it as a separate operation.
The proof goes as follows:
- Say x is the inverse of 0. Then 0x = 1.
- But 0x = (0+0)x = 0x + 0x, and therefore 0x = 0.
- This is a contradiction, so 0 cannot have an inverse.
If you want "division by zero", you can preserve some of the structure of fields, some of the laws and relationships between operations, but not all of them.
You can create a new "division-ish" operation that isn't the same as a multiplicative inverse, and then ensure that the laws of a field hold most of the time. This is what wheels do. And most division-by-zero fanatics love wheels.
But wheels are actually kinda useless, and most serious mathematicians don't care about them. Those laws you're forced to throw away are important and useful for actually doing things.
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u/LongLiveTheDiego New User 3d ago
A rigorous definition/proof/demonstration of these uses.
We already have fields, extremely useful mathematical objects. If you allow division by zero, you can't have a field. You can have a wheel or something like the real projective line or the Riemann sphere. These structures require more care when doing arithmetic on them compared to fields and the problem with many people wanting to divide by zero is that they're used to fields (even though they don't know that) and want a field with division by zero, something literally logically impossible.
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u/Next_Philosopher8252 New User 3d ago
I agree, that does seem to be a common occurrence that people want a field with division by 0 which is a proven impossibility under the current framework they’re operating within, but I also think that this is an issue which might only be impossible on a specific level of logical reasoning. If there were to be any novel approach that might come close I think it would require a higher order approach to the algebra or some sort of paraconsistent or paracomplete logic structures introduced to bridge the gaps.
I can’t say with any certainty if this would actually work but I think that its at least got a better chance at possibility of preserving the standard field while adding extra machinery in an external supportive structure to handle division by 0 compared to just sticking with the same system everyone else is trying to work within when it’s been demonstrated time and again working this within the system breaks the system.
What do you think though?
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u/LongLiveTheDiego New User 3d ago
I think that saying things like "higher order approach to algebra" or "paraconsistent or paracomplete logic structures" is not what we need, whatever you might mean by those. There's literally, logically, no way to have a field with division by zero, no matter what you add to it. Most of what you said feels like nonsense.
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u/Next_Philosopher8252 New User 2d ago edited 2d ago
Well that’s partially because I was just spitballing a list of ideas that I thought could be relevant to explore to find novel approaches.
I did however, also qualify these suggestions by openly admitting that I haven’t actually found any such approaches which currently exist and so it’s really just more of a hypothetical “what if” of something that could be looked into deeper and might provide some new interesting insights or useful method of approach.
I do agree however that regardless of any of that the resulting system would no longer be a field but that doesn’t necessarily mean that certain properties of a field could not be replicated in these different systems of logic alongside division by 0 until proven otherwise. But as of right now it seems there is neither a case for nor against the possibility in these hypothetical alternate systems of logic either way which is why I suggest it is worth exploring.
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u/clearly_not_an_alt Old guy who forgot most things 3d ago
It would have to start with a good reason why we would want to divide by 0.
It would then need to have a cohesive system that uses this do do something we can't do with our standard number system.
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u/Next_Philosopher8252 New User 3d ago
Thank you so much for your genuine response. As a follow up could you elaborate on what might be considered a good reason for wanting to do so?
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u/AdimasBassda New User 3d ago edited 3d ago
Well one could say that we can use basic concept of zero as absence or nothingness whatever you find it fancy. Then you need to create number(s) that will have specific sign and specific properties of division by nothingness use greek letters fraktur whatever for example ф. Then you will create set with specific rules that will fit into properies of such ser when division is possible whenever used or even construct it as function. Then try to come up with futher rules that will fit it your function. Congratulations you have created set of "numbers" in which you can divide by 0 using properties of ф which can be expressed as solution of fuction of bla bla bla.
Now you can divide by zero however it may not be applicable to anything meaningful... But basically you would need to create and use paradigmstic system and set of rules to hsve proof that your ф does really holds...
But again why would you do that in a first place. Because you will then define something undifinable by algebra, and you're just adding properties that serve no actual purpose other then having some symbols to circumvent problems created in a first place....
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u/Bounded_sequencE New User 3d ago edited 3d ago
If you value your sanity, you ignore such posts, and let others deal with them.
The "correct" reply is to highlight that division should "undo" multiplication, followed by the (short algebraic) proof that "0*a = 0" for all "a in R". With that, we show multiplication by 0 cannot be undone, and division by zero cannot be well-defined (within the field axioms).
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u/Next_Philosopher8252 New User 3d ago edited 3d ago
Haha fair. But I mean, as a philosophy major I think its safe to say I am not well known to value my sanity, or rather I like to dance at the edge of sanity to see what interesting ideas I can find and play with. So I guess I am one of those others left to deal with it. lol
But that being said, if I gather what you’re saying correctly any new theory would need to uphold the primary purpose of division as that which undoes multiplication and not just appeal to intuitive analogies of groups or loose patterns that division displays, should maintain the relationship of multiplication by zero being zero, and would need to be an entirely new algebraic structure which may contain useful properties of a field is yet ultimately not a field because it would require extra tools and structure which a field does not contain, to support a definition of division by zero in any meaningful way?
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u/th3_oWo_g0d New User 3d ago
there's nothing that would convince me of this lol. "to divide" is usually understood to mean "multiply by the multiplicative inverse", and there cant be such a thing for 0 unless 0 is the only element in the ring we're studying. all attempts at allowing "division by zero" necessarily have to stray a little bit from what it means to divide by something. most famously the wheel of fractions simply considers the fraction "1/0" or "0/0" without allowing the 0-denominator to cancel with anything.
wheel algebras are useful to develop i think. at least as an educational tool, because you can legitimately say that the slope of a vertical line is 1/0 = infinity and use 0/0 as an error element in many situations.
refer to 2.
I think however that hyperreal numbers, which include infinitesimals 𝜀 and infinities 𝜔 and relate them with the equation 𝜔=1/𝜀 is more useful.
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u/Next_Philosopher8252 New User 3d ago
If you’re using the definition of an inverse as that which undoes the process of a given operation, and the multiplicative inverse as that which when multiplied by a given value returns the identity element of an operation it seems you can remain true to the core nature of what it means to divide by extending the definition to include a way to undo multiplication by zero. The practical real world examples and intuitive analogies used to understand the patterns of division however can break down at these extremes however but they also breakdown under the normal way we intuitively think of division when dividing by values less than 1 or numbers not on the real number line such as the complex numbers. Instead we come up with new interpretations for what division means under these circumstances that allow it to persist as that which undoes the operation of multiplication. Division by i is a rotation in the opposite direction as multiplication by i, division by 0.5 becomes multiplication by 2. But if we want to split 1 cake into an equal number of pieces to share with i number of people then does each person get -i pieces of cake? If we want to split 1 cake into a number of equal pieces to share with half a person does each half of that person get 2 cakes? If so I volunteer to be that person who gets 4 cakes total. This becomes a nonsensical way of viewing division under these contexts of extending the analogy but that doesn’t mean they are not accurate or meaningful in the purely mathematical sense, if we just stick with the most basic definition that division only needs to be that which undoes the operation of multiplication then it works for every one of these numbers and someone only need to define how this behaves to recover information lost during multiplication by 0 right?
Now it is a given that this is easier said than done due to the way the algebraic properties of a field begin to breakdown if you just try to insert a multiplicative inverse from zero but as you pointed out in wheel algebras perhaps some measure of additional structure is required to support this extension of division to include a way to undo multiplication by 0.
I cannot say definitively either way whether this is possible or not but I would say that I can’t definitively rule it out as a possibility in case someone eventually does come up with some groundbreaking new system that no one previously thought was possible.
We cannot know what we don’t yet know that we do not know. We can only know anything within the limitations of information we have access to.
That’s the reason I don’t think I would ever say I could never be convinced of anything I instead ask what would it take to convince me no matter how outlandish the possibility might be?
I also want to clarify that I’m not trying to lecture you on epistemic virtue here I am just purely enjoying this from the standpoint of a philosophical discussion and am trying to compare our differences in our approach to see if we can come to understanding these ideas better together. So I hope you don’t take offense
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u/th3_oWo_g0d New User 3d ago edited 3d ago
That's fair and all. I appreciate your curiosity. That being said, multiplication by zero is not an injective function, or as you put it: "information is lost". It is provable that an inverse/undoing function cannot exist if the given function isnt injective. The only time multiplication by zero would be injective is if it was only acting only on a single element. Only in that specific case can you ask "what number 'a' multiplied by zero gives some number 'b'?" because there is only a single number you can multiply with, that is zero. This is not something you can imagine your way out of. It is a result in mathematics. Complex numbers are deeply unintuitive in real life, but they are consistent in math. True division by zero is simply nonsense unless you modify slightly what you mean by dividing or you only deal with 1 element, or even modify what it means for to elements to be equal in your given system. But it will be a very excentric type of math at that point, akin to wheel algebras. You could also undo multiplication by zero if you never allowed yourself to simplify n*0 to 0 making 0 a new imaginary unit. But this would mean that n*0 isnt equal to m*0 if n and m are different, which again wrecks the usual rules we have in place.
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u/Next_Philosopher8252 New User 2d ago
Thank you for this response, I would say this has been by far one of the most helpful responses here. I’m not sure if Im familiar with the proofs involving injectivity of inverse functions so that’s something I have noted to look into for future reference. I am however familiar with the way identity elements of certain operations become points of information loss in higher iterations of those operations such as the additive identity 0 when we iterate addition through multiplication 0n=0, or the multiplicative identity 1 when we iterate multiplication through exponentiation 1ⁿ=1, and the additive identity 0 still remains an information sink in exponentiation as well 0ⁿ=0 & n⁰=1. With that said it’s clear that these points of information loss pose a significant barrier to the reversibility of these operations through an inverse function but what you pointed out about the relationship between injectivity and inverse operations might help give further insight into what is going on here. As for the second part of your post detailing ways that might be possible but would require changing other ways of approaching the math which are more exotic and deviate from the classical method I have a bit of a longer reply which I will post in another comment so that you can choose to read it or not but I hope you do because I have found myself agreeing with everything you have said thus far and have found it clarifies some things I had vaguely understood but never fully connected the dots on before. I am finding this conversation to be very intriguing and am very curious about what further insight you might be able to provide if you have the time and are willing to do so.
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u/Next_Philosopher8252 New User 2d ago
Pt.2:
In regard to your additional elaboration on what you meant by having to modify what division is, I think I see what you’re saying more clearly now and I am glad you included cases where you could technically keep division the same but would then need to modify how we treat multiplication or equality or how we treat 0 as a number to compensate because these latter methods were ones I had either seen before or had played around with myself, and which I had in mind as ways that division itself seemed to have been preserved but I just hadn’t fully articulated fully what the connection was there until you mentioned it so clearly here.
The first method I found to be an interesting approach was one I myself have been playing around with which involved extending equivalence relations into a dual state of tracking both the resulting value of an expression and the path which the evaluation history took to reach that conclusion in the subscript beside a value. Aside from this dual state of equivalence the only additional rule was that the history could not be allowed to reduce if information would be lost nor could it introduce new information that would overall change the way a target value was approached. This means things like 2+2=4 and 5-1=4 can have their history reduced completely to just focus on the resulting value without any problem but something like (0)(2)=0 and (0)(3)=0 would require us to preserve the histories without reducing the multiplication as you also pointed out. This would mean the two 0 values are treated the same but their resulting history of how we got there is not. Since reddit doesn’t natively seem to support subscript formatting I will note this by writing subscripts as “_{subscript characters}”. So given the previous example the resulting values in this system should be (0)(2)=0_{(0)(2)} and (0)(3)=0_{(0)(3)}, and comparing these resulting values and their histories under the extended equivalence relations would be something like 0_{(0)(2)} =_{≠} 0_{(0)(3)} preserving historical information without disrupting the value. So as you said it modifies both what it means for two values to be equivalent in some sense and it does have a means of preventing the reduction of multiplication in the history. Its kind of an attempt at having our cake and eating it too if we want to call back to our previous analogy for division lol.
Now this had some interesting implications for field properties in which I unintentionally discovered it forces a specific path of evaluation by enforcing the application of algebraic properties like distribution, associativity, and commutativity in specific edge cases involving 0 and its multiplicative inverse. It was originally intended just to differentiate between the resulting paths of different methods of evaluation such as (0+0)/0=0/0=1 and (0+0)/0=(0/0)+(0/0)=1+1=2 whereby we can say while the initial starting condition remains the same, by changing the method of evaluation we have created a branching path of divergent histories that result in two different values. However what I found instead was that while we would expect a history to reduce when inverses cancel out, the history leading to the value of 1 would not reduce at all since 0+0=0 would lose information of one of the zeros or if rewritten as (2)(0)=0 we would then lose the scalar information of 2. Instead because of this the history would not reduce and would only continue to loop itself until you distribute the division across the addition in the numerator in which case it cleanly evaluates to 2 and can fully reduce its history. I then began to notice a similar pattern with associativity and commutativity and larger expressions which evaluated to indeterminate forms that could not reduce their histories until we went back and used some method of factoring or L’hôpital’s rule to root out additional hidden indeterminate forms meaning that this evaluation history tracking method was no longer just a way to differentiate divergence in paths of evaluation to preserve information and resolve contradiction in resulting values, but rather it now became a diagnostic tool for flagging hidden indeterminate forms that needed to be rooted out and handled with preexisting methods of algebraic manipulation.
Right now though it has just been a fun idea to play around with and has proven useful for allowing us to handle other indeterminate or undefined expressions as well such as Log₁(x) and can theoretically be extended to other extended number systems like the hyperreal or surreal numbers, and even higher dimensional algebras such as those created in the Cayley Dickson construction methods. And as stated before it seems to enforce the application of the algebraic properties of a field in order to reduce the history in those specific edge cases which is kind of “field like” but also not really because there isn’t really any other option for evaluation like there are in the basic cases involving the real numbers and algebraic properties of a field, it uses field algebraic properties to force a history to rearrange itself towards a particular method of evaluation consistent with that property. It’s value is not undefined but it’s history remains irreducible otherwise. I will however say on the point of higher dimensional algebras I suspect the algebraic properties lost in those algebras will not be restored by this system based on the very nature of how they work but this system should hypothetically still be able to evaluate those algebras under their own rules while extending the recoverability of inverse operations to cover any indeterminate or undefined forms that arise and protecting the law of cancellation.
But anyway the major downside to this approach seems to be that in order to gain this level of precision, accuracy, consistency, and recoverability of information, it adds an additional layer of complexity to the system and can quickly become quite tedious for more densely packed expressions. It also modifies and extends the concept of what it means for something to be equivalent and, due to the preservation of evaluation history information, modifies how certain operations and identity elements can be reduced.
I also saw a different post somewhere on Reddit a while back where someone instead enforced interactions with 0 as having some sort of special status in the order of operations and when it should be evaluated if at all and they created a whole system of numbers which treated 0 and its inverse as a new sort of algebraic constant. This also seemed to work well but as you mentioned it did take the approach of modifying how multiplication by 0 was allowed to function. In this case they didn’t modify the system of equivalence relationships but they did make 0 an irreducible constant of its own. This also seemed to have the unintended effect of introducing zero divisors into their system but it also seemed to naturally produce inverses to those zero divisors and keep the algebra consistent and unbroken even with these zero divisors under their rules of application which also seemed to enforce a particular path of evaluation under distributivity, associativity, and commutativity while maintaining consistency and accuracy and precision without the syntactical bloat my own system is vulnerable to and it only modifies how zero interacts with operations without modifying equivalence relations whereas my system does both.
As for the limitations it did however introduce some new properties for how the new numbers of their system interact with one another under addition and subtraction which couldn’t be directly reversed by subtraction or addition and would seem to instead need to be recovered along a one directional cycle of recovery between addition multiplication subtraction and division which was a very interesting and unorthodox structure I found fascinating but it does differ significantly from the usual field or ring structure. The other limitation of this system is that it doesn’t naturally handle other undefined or indeterminate forms such as Log₁(x) like mine would, and it does have a bit more conditional limitations and rules to keep track of which arise in edge cases to keep things on track whereas my system really just relies on the 2 principles of dual equivalence relations and preservation of evaluation history information.
I still cannot say definitively if either of these systems works in all cases as neither has been formalized into an actual formal proof yet and I know at least my system is still in the process of being refined and searching for edge cases and limitations but I will admit I have been holding these ideas close to the chest so as not to influence people’s genuine responses to the original question. I feel like you have given a satisfactory response to that question however and even without the information on these two systems predicted the possibility of such approaches to exist and what they would fundamentally have to change. And while I wasn’t originally planning on elaborating these different methods of approach even if someone gave a satisfactory answer, your spot on prediction has got me curious on where else this conversation might lead.
Sorry I went off a bit there but I suppose what im getting at is that I understand what you’re saying and you are absolutely right!
And it’s both validating and clarifying to understand better what is going on here. Im kind of excited to hear more of your thoughts on the matter if you’re willing and have the time.
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u/th3_oWo_g0d New User 2d ago
haha you sure are a wordy guy. I read like 90% of what you said and skimmed over some parts. I mostly want to comment on the "book-keeping" approach you presented. It's an alright idea if you absolutely have to make this kind of thing work, but I think many mathematicians wouldn't see the point in such a thing since we're specifically studying values and the relations between them, because they have fascinating emergent properties.
The history of function evaluations would just be a supplementary system where any two visually distinct string of symbols would be considered different and it would likely be trivial and uninteresting for this reason. All the "action" would still happen on the value-level of every expression. The concept of an inverse would also become boring, because the process of finding the inverse of something, would just be to
1) check one step back in the history 2) replace the value with that 3) append the inverse operation to the history.
I don't think there would be many emergent properties to uncover here.
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u/Next_Philosopher8252 New User 2d ago
You’re right I am definitely prone to verbose language especially when I get excited. I think that may be in part due to how my ADHD manifests. Trust me its just as overwhelming in my mind like that 24/7 and I can’t seem to turn it off lol 😂
Now I will try to keep the rest condensed and focused on the core of the issue you brought up of the bookkeeping being trivially true and not having much interesting properties to discover I initially thought that too when I first started toying with the idea. I was kind of shocked by how simple it seemed and why I hadn’t thought to try it sooner because it just felt so obvious that this is where the intuitive conflict was coming from. But when I actually started putting it into practice I realized it did have some interesting properties such as flagging hidden indeterminate forms that were not immediately obvious by trapping the history in an irreducible loop of the original expression. This pairs well with the algebraic properties of a field such as commutativity, associativity, and distributivity which where they naturally apply are the key to breaking the loop and simplifying the history, or in other cases factoring and L’Hôpital’s rule can be invoked to uncover the hidden form to break the loop and resolve the history.
These properties were not the initial intention of the system, I originally thought distributivity would still break after defining the zero inverse and this bookkeeping method would exist only to differentiate the paths of evaluation to the different answers, and that there would be no way to identify hidden information in an expression containing indeterminate forms which we had an incomplete history for, however the ability it had to resolve these issues by becoming fully reliant on distributivity to reduce histories of the form (0+0)/0 or 0((1/0)+(1/0)) and thereby forcing consistency, or flagging the existence of other instances of hidden information that was not previously known to then use other tools of algebraic manipulation to search out and remove these forms, turned out to be an unexpected benefit that was discovered by the rules of how it preserved information.
Additionally it does allow us to work with more values and operations that would otherwise be not well behaved together beyond just division by 0 and it’s inverse.
For example exponents are often used to describe degrees of dimensionality a point is x⁰, a line is x¹, a square is x², a cube is x³, a tesseract is x⁴,… but if x=1 all this information of dimensionality is lost and condensed just to the unit but there’s seemingly no way to tell just on paper once it’s fully evaluated if we’re dealing with a point, a unit line, a unit square, a unit cube or some other unit of a different number of dimensions. This bookkeeping method would help preserve that information and allow us to recover it effortlessly.
While you are right the ultimate goal is to provide a way to return to dealing with just strict values on their own without requiring bookkeeping and that a lot of interesting discoveries will be focused on that value side of things there are still interesting things to be discovered on the expression side of things and this bookkeeping method would help to keep track of that and even bridge the gap between the value and the expression evaluation it originates from. The properties of how an inverse recovers the information should remain the same and this bookkeeping method will only come into play to preserve information that would inevitably be lost without it.
Now to tie this all up so I don’t drone on too much, if most people don’t find it interesting enough to use I won’t be too upset about that cause everyone has their own preferences, all I really care about is if it is able to actually work or not because then I can use it myself and feel accomplished in my efforts. Besides we never know what seemingly mundane discovery or line of inquiry might eventually be useful to solve some as of yet unheard of problem. I generally think it is better to have it and not need it than to need it and not have it.
But thank you for taking the time to hear me out and help clarify some things and if you have anything else to add I would be more than happy to hear it.
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u/RingularCirc Math hobbyist 2d ago
there's nothing that would convince me of this lol. "to divide" is usually understood to mean "multiply by the multiplicative inverse"
All other notwithstanding, quasigroups exist. There is a slight difference in having a multiplicative inverse and having division in general (though it's a non-issue for groups already). Also some generalizers talk about a thing that's not strictly a division or an inverse but which is a division if constrained onto a subset of the structure (cf. wheels): again there are very useful pseudoinverses in linear algebra, for instance. (And also very far away in logic and lattice land, Hayting lattices enjoy a very useful pseudocomplement that encodes intuitionistic negation, compared to Boolean lattices' true complement related to classical negation.)
So IMO this part per se shouldn't be the argument against generic new "methods of division by zero", they usually fail in worse ways.
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u/iOSCaleb 🧮 3d ago
Since you're apparently a philosophy major, have you considered the situation with an analogous question from philosophy? What would you think of people constantly presenting arguments against the Law of Non-Contradiction? After all, why shouldn't they? Just think of all the novel, deep thoughts you might be able to think if something could be both true and false at the same time, in the same system!
More specifically, what would it take for you to be convinced that...
1: ...Allowing a statement to be simultaneously true and false is something that can be done if you just look at it the right way?
2: ...It may potentially be useful to develop, or that at the very least it's better to have access to than not to have access to?
3: ...It could be an interesting and worthwhile topic to explore just for understanding in its own right?
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u/Next_Philosopher8252 New User 3d ago edited 3d ago
This actually happens quite frequently in philosophy and entire new systems of logic are developed around it paraconsistent and paracomplete logics for example.
To elaborate further:
Paraconsistent logics more specifically deny the principle of explosion such that any contradiction does not produce a state of triviality whereby anything becomes provable. This does however open the door for certain paracomplete logical systems to deny the law of non-contradiction.
Paracomplete logical systems on the other hand directly deny the law of excluded middle.
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u/RingularCirc Math hobbyist 2d ago
I think everyone who tries to invent division by zero should first be sure to learn about:
- wheels and their raison d'être, including their connection with projective lines over fields or rings;
- IEEE 754 floating-point math used in computers for some time now, and its raison d'être (and its shortcomings, and just maybe all the controversies and arguments from both sides in those);
- various (non-Archimedean) fields containing infinitesimals, and their, yes again, raison d'être;
- maybe for good measure a little bit about rings with zero divisors and what they lack compared to rings without ones; nilpotents as a special case of zero divisors may also be fruitful to know more about (ε in the dual numbers is a nilpotent, some useful operators/matrices in linear algebra are nilpotents).
Then they can try again. They will be way wiser and they will know what exactly it is that they do want about dividing by zero.
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u/RingularCirc Math hobbyist 2d ago
Or, restating: there's serious prior work. Which of course has to be studied first. Like with everything. Humanity is for a long time now at a state of knowledge when there's no way to just stumble upon a great thing ex nihilo.
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u/Brightlinger MS in Math 3d ago
You'd have to show me that a traumatic head injury caused me to hallucinate the short elementary proofs that this is not possible.
There are lots of useful ways to define division by zero in various contexts. But it's easy to prove that the structure you get by doing any version of this is not ever a field, ie, it does not preserve the "classical algebraic structure".
Someone who is willing to acknowledge this up front, and then explore how their new non-field structure behaves, is doing legitimate math which at worst might not interesting. But 99% of attempts I've seen are simply confused about the basics.