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Pondering a Proof
today, i overheard my nerd friends (alright, my friends =.=') that the proof for 1*1=1 does not exist. So I'm just curious, how do you prove that 1*1=1? Is this just an intuitive fact that us mathematicians as true? Is it something like questioning the fundamentals of time, a human invention just as math is?
discuss :D |
Re: Pondering a Proof
Unless 1 is just a variable, I don't really see how it can't be 1.
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1 group of 1 unit = 1 unit...makes sense to me.
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I did some googling and found the proof for 1 + 1 = 2
It looks quite complicated with fancy postulates that i don't understand: http://mathforum.org/library/drmath/view/51551.html i was expecting for something like that...but with 1*1 = 1 -edit- lolz i found a video of it http://video.google.com/videosearch?...um=5&ct=title# oh wait, it's 1 + 1 = 0...and it's flawed. *sigh* googling attempts foiled yet again by misleading titles! -edit- Quote:
It's like sex. :) |
Re: Pondering a Proof
Wouldn't it just be:
1*1=1 1=1 |
Re: Pondering a Proof
well for 1 + 1 = 2...
the same logic is as you said 1 = 1 YET THERE HAS BEEN A PROOF DERIVED |
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There are postulates out there. I'm guessing if there isn't a proof for 1 X 1 it's just a postulate.
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Re: Pondering a Proof
So, a lot of mathematics really is just built on a set of axioms that we deem is true in some system. In most cases in some structure, the existence of some "unit" 1 such that 1*g = g for any other element g in this structure is one of the given axioms. There really isn't much to say about this - it's just something that's assumed to be true and which allows us to study structures without having to really deal with what this unit "1" actually means.
In terms of a proof, it really goes down to breaking down mathematics in terms of whether or not certain axioms are correct or not, and whether or not the mathematical system we've developed is consistent in all cases. This gets VERY, VERY convoluted, which is why the proof to 1+1=2 is quite long, since all of mathematics essentially has to be dissected and put back together. If you're interested in stuff like this, definitely look out for a class in Mathematical Logic when you go to college, or some equivalent. There you'll formally break down what mathematical systems mean and prove a number of important results that will help you get a firmer grasp on what all these systems actually represent. |
Re: Pondering a Proof
In the group of multiplication with real numbers, 1 is considered the identity element. Therefore for all n, n * 1 = n. Granted even that proof doesn't dig into why 1 is the identity element, but we're getting closer.
The word "axiomatic" is sometimes meant to mean, "taken for granted." Basically, we take axioms in math for granted. As far as 1 * 1 = 1 or anything like that, you don't need a proof. It's obvious. We can take it for granted. |
Re: Pondering a Proof
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So yes, on the surface this seems like a question that requires a one line answer - "well, of COURSE 1 * 1 = 1, that's how we defined it!" But on a deeper level this is a very important issue that has plagued mathematicians, and more specifically logicians, since the establishment of mathematics as a rigorous system. I think it boils down to - is there really any reason to have complete faith in mathematics as an absolute system? |
Re: Pondering a Proof
"I think it boils down to - is there really any reason to have complete faith in mathematics as an absolute system?"
Yes, if and until if becomes broken. It becomes broken when using it wrongly describes some aspect of the universe. If it doesn't describe it properly, it simply hasn't been 'invented' yet. |
Re: Pondering a Proof
Let's try it with x*1.
x*1 = x Why? Because it's one of x. 1x*1 = 1x Anything that exists tangibly or theoretically exists in the form of 1, possibly manipulated by an action or equation. So if x = x then 1x = 1x and 1 = 1. Any one of something is that same something. It's one of it. This reasoning implies that any negative number has a positive counterpart though, as something must exist before it can unexist. Physically speaking though, it always existed, just in a different form. I'm confusing myself. xD Good luck. |
Re: Pondering a Proof
1+1=2
1(1)=1 I know this is CT, but why are you debating over something so clear and obvious? I am not really what you would call 'mathematically oriented', but this is definitely some logic I have never pondered.. |
Re: Pondering a Proof
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peace |
Re: Pondering a Proof
1 times 1 times 1 times 1 times 1 = 1.
It's like an infinite loop!!11!!1!one!!11! |
Re: Pondering a Proof
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Math doesn't necessarily become broken just when using it incorrectly. Sure, that's a possibility, but we must ask ourselves if axiomatic mathematics is justified in the first place. With a set of bad axioms, even when using it correctly, you can prove a lot of really stupid things that are obviously false. If you want, you can google the proof that Winston Churchill is a carrot, just based on the stupid initial assumption that 1=2. Perhaps I'm just misinterpreting your statement... I'm not quite sure what you mean by "describing some aspect of the universe." Of course, this brings up a whole other issue of: is math actually physical? Sure, we can apply math to physical systems, but aren't these but shoddy impure tangible copies of what we think of as the ideal mathematical objects? You can look at a basketball and think of a sphere, but really a sphere is an idealized mathematical object that does not really exist in nature. This brings up the issue of - what about a basketball allows us to invariably make the connection between it and a sphere? At this point, I may be getting off track... so I think I'll stop while I'm ahead. The truth, and most of the posts in this topic confirm this, is that almost everybody takes math for granted. Oh sure, it's obvious that 1*1=1. But is it really? Not when you really think about it. I'm pretty sure if you were to grow up being taught that 1*1=2, and building mathematical models off of that fact, then when faced with the question "why is 1*1=2?" you would most likely scoff and go "well, it's obvious, isn't it?" Unfortunately, it's not that simple. |
Re: Pondering a Proof
it's obvious that 1*1=1 because it's part of the definition of the integers
math is axiomatic, it works the way it does because we make the rules if 1*1 equaled 2 then it would be a different system god |
Re: Pondering a Proof
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Well, then, what exactly are the integers? In terms of naive set theory, there's a very rigorous process by which the integers are constructed. So, really, we can say that 1*1=1 if 1 is considered to be an integer because of the axioms of set theory. Of course, then we must wonder whether or not the axioms of set theory hold in any case. Do we really have any intuition as to what a "set" actually is? Can we just assume the existence of sets? So no, unfortunately you're making a gross oversimplification. In fact, you're pretty much arguing in a circle - "1*1=1 works because we defined it that way. We defined it that way because it works." But no, how really do we know that it works? This is really a more serious issue than most people who have posted in this thread believe... |
Re: Pondering a Proof
The problem is that if you're only concerned with the actual statement 1*1=1, the simple expression of that as "one group of objects wherein each group contains one object is a set that contains one total object across all groups" is obvious not because of the nature of integers, not because of the way set theory works, but because of the lingusitic definitions of the words "one" and "group"
The only way this is complicated is if you somehow want to question the linguistic definitions involved. |
Re: Pondering a Proof
I hate questions like this because no matter what, you're going off SOME sort of assumption about how things relate. 1*1=1 because you have one 1. Easy. I mean, why do you need a "proof" of this? It's like saying I am holding one apple. Prove I have one apple. By definition 1*1=1! I never saw the point in these kind of questions.
As for those "proofs" showing that 1=something other than 1, those have never impressed me because every single one basically plays off mathematical ignorance of the majority by violating some fundamental step along the way or making some assumption that is clearly, in itself, not a valid assumption. |
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And MrRubix, you can legitimately say that 1+1=1 if you define what you mean by "1." If, for example, I use 1 to refer to an equivalence class of integers modulo 1, then it is correct to say that 0=1=2=3.... etc. Of course, if we are purely working in the system of integers, then such a statement is impossible, but then again we have to axiomatically define the integers. And it certainly is not rigorously obvious to jump from the mathematical concept of "1" to having "one apple." Once again, it's important to treat "1" as a purely mathematical object, or else this thread really doesn't have a point. We're not allowed to assume that "1" has any physical meaning attached to it unless we actually show that such a physical meaning is justified. As of yet, nobody has really posted anything without making a boatload of assumptions... This really isn't an issue we can resolve within the space of one forum thread (indeed, it's been a hot topic in mathematics for much, much longer than this thread has been running). Still, I feel it's important to see what the issue is here. It's not a matter of linguistics, or physical intuition, but rather a matter of mathematics as a pure system, and whether or not that system can be defeated at ANY angle. PS: Linguistics really isn't a great way to study mathematics. In fact, the English language has so many ambiguities in it that simply using the English language to define mathematical concepts is completely unrigorous. Take the following example, for instance (which I have taken from Munkres, Toplogy, 2nd Ed): Compare the following two statements: (1) "Miss Smith, if any student registered for this course has not taken a course in linear algebra, then he has taken a course in analysis." (2) "Mr. Jones, if you get a grade below 70 on the final, you are going to flunk this course." In statement (1), the logical flow is that if student A has not taken linear algebra, then student A has taken analysis. However, if student A has taken linear algebra, then he may or may not have taken a course in analysis. In statement (2), the logical flow is that if Jones receives a grade less than 70, he will flunk the course. However, it is understood from context that if he does not get a grade below 70 on the final, he will not flunk the course. This is, in fact, the converse of the statement. In other words, statement (1) reads "if P, then Q", whereas statement (2), albeit it also reads "if P, then Q," is understood to mean "if and only if P, then Q." Mathematics does not allow this. If a statement reads "If P, then Q," then the converse (if Q, then P) of the statement definitely does not have to hold true. The bottom line is that we should really not try to analyze mathematics in the context of linguistics. There are a lot of logical problems about the English language, and many of them need to be fixed before we can even try to make sense of math. |
Re: Pondering a Proof
Rather than try to analyze math in the context of linguistics, I was instead suggesting that his question was one of linguistics to begin with, and not of mathematics.
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Re: Pondering a Proof
Well, I was referring more to the part where the original poster mentioned that his friends had claimed that the proof for 1*1=1 does not exist. That claim probably comes from a mathematical perspective, since obviously, 1*1=1 holds if we define 1 linguistically... it doesn't really need mentioning.
So sure, I concede that if we were talking linguistically, there's not much to say. But I don't think that's the case, and there remains a formidable problem if we're staying in the realm of mathematics. And plus, it's more fun this way anyways =p. |
Re: Pondering a Proof
Seeing as we're not automatons, we're never purely staying in the world of mathematics. The only way someone could understand math purely logically, they'd need to have no sensory input whatsoever, in which case I do not think thought for them would be possible. I am NOT saying that we cannot abstract things beyond our senses. I am saying that the development of our brain would be non-existant. As such, we're innately tied into our senses, and there's no way out of that.
My understanding of math is that it is designed to explain the world around us. If it fails to do so, then obviously there's a problem with it. If 1*1=1 is not mathematically provable to us, then that would be a problem with the system of math we've designed it seems, since our math system revolves around proof. However, that we've designed a million physical things that have used the rule that 1*1=1, I'm confident that it is correct. Actually, that you say it's up to math to prove 1*1=1 and not language seems short-sighted. What exactly are the symbols "1*1=1" if not language? A mathematical proof that does not lend itself to being shared in our physical understanding of the world is a non-existant proof. |
Re: Pondering a Proof
Well, math is certainly a language.
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I'm not sure of the details of a 1x1 = 1 proof, but under the assumption that it cannot be proved (likely not the case), it's an example of something that is self evidently true that cannot be proved within the formal system of mathematics. As many of you have shown through simple logic, it is not hard to see that 1x1 = 1 is true. Because of the axiomatic structure of mathematics however, that doesn't necessarily mean it's easy to prove. This isn't surprising. It's a problem of regression; the system will always be incomplete because you're never going to have all of the axioms, thus leaving some portion of the formal system unprovable. However, thankfully axioms are, generally, obviously true, so I don't think we have a real problem here. |
Re: Pondering a Proof
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It is definitely possible to think logically, and purely logically, and it's something that mathematicians have been doing for a long time. And yes, 1*1=1 is, in fact, use of language to define a certain statement. However, this is NOT English. Rather, mathematicians work more predominantly in "First Order Logic," a mathematical language that uses completely unambiguous language to a satisfactory level of precision. This is not English, and as I mentioned before, it's not very conducive to rigor to think about mathematics in terms of English. 1*1=1 is not a proof in the conventional sense. 1*1=1 is a foundational axiom that most mathematicians deem to be true, and have built a lot of mathematics around. That is not the issue. The issue is whether or not it works as an axiom in the system of mathematics. I know that at this point, I'm probably getting repetitive in my posts, but almost everybody has been urging use of a physical analogue of math to intuitively prove 1*1=1, and I'm claiming that such physical analogues are irrelevant. The physical analogues are relevant only as a corollary after we've proven that 1*1=1 works in a consistent system. "Reach" probably comes closest to this realization - because 1*1=1 is an axiom, it cannot be proven within the system in which it rests. That would perhaps be the biggest piece of circular reasoning ever - we build a system off of this axiom, and then use that system to prove that this axiom is true. Obviously, this does not work. Instead, what I'm suggesting is that we look at 1*1=1 as an axiom of a system, and then look at this system externally to see whether or not it is coherent. Words like "confidence" and "intuitively true" have very little meaning in mathematics. They may help you determine the best way in showing some result, but they are never sufficient by themselves to show a result. Ultimately, to me it seems that we will never be able to prove everything in mathematics. To prove anything, we need to first make assumptions, based on the system we are in. The question is, what is the absolute simplest set of assumptions that seem "intuitively true" that we can make? Obviously, 1*1=1 is not the absolute simplest thing we can say. There are much simpler things that we can build 1*1=1 off of. Furthermore, we have to define what "1" is, what "*" is, and what "=" is in terms of elements of this language of logic (what "=" means is quite another topic altogether that would probably need the creation of another thread to explore). It becomes a messy business, but ultimately it's rewarding to assume the least possible and still come out with the correct results. |
Re: Pondering a Proof
If you think of(define?) addition as repeated multiplication, then it makes sense.
EG: x+x+x+x = 4x 2+2+2+2 = 2*4 = 8 1+1+1+1 = 1*4 = 4 1+1+1 = 1*3 = 3 1+1 = 1*2 = 2 1 = 1*1 = 1 I'm guessing this doesn't work for some reason because otherwise someone else would have thought of it, but oh well. I'm still interested to see why it is wrong, if anyone knows. |
Re: Pondering a Proof
well 1*1=1 so divide over the 1 it would be 1=1/1 umm its pretty simple...
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Re: Pondering a Proof
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You can prove axioms, but in order to do it you have to fabricate further axioms. I think Euclidian geometry is a good example. You need to formulate several axioms in order to complete the system, but in doing so you can no longer prove the axioms (although they are likely true simply from observation) unless you plan to fabricate additional unprovable axioms to prove them. As such, I don't think a complete formal system is important. If the portions of the system which cannot be proven remain obvious through simple perception, then mathematics can adequately explain the perceived world. I don't see the point of describing anything else >_> |
Re: Pondering a Proof
"As such, I don't think a complete formal system is important. If the portions of the system which cannot be proven remain obvious through simple perception, then mathematics can adequately explain the perceived world. I don't see the point of describing anything else >_>"
This. Anyways, when it comes to math, I'm perfectly content with letting the obvious things be "objectively true." I'll save the "potential fallacies" for physics. |
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i said how the math works... what don't u get?
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There are a lot of things that are not precise if you want to structure a proof like that. Be careful. Sidenote: Actually, on second thought, did you really say anything new? You essentially reformulated the proof of "1*1=1" into a proof of "1/1=1" (of course, assuming that these operations are all well define). If "1*1=1" is not obvious (as we are assuming for the purpose of discussion), then "1/1=1" is definitely not more obvious. So... what's the point here? We seem to be no further than where we had started. |
Re: Pondering a Proof
after further thought i do see what u mean. with my "theory" it is saying the this could also equal 1. 1/1*1/1 but that would equal 2. when u multiply fractions u have to add across. so i see where the debate comes in. now i really want to know how this makes sense. its this it like Einstein's theories. if you cannot prove them wrong then they must have something in the theory that is one the right track. im pretty sure its just and accept thing as u said before. >.>
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Re: Pondering a Proof
"It is definitely possible to think logically, and purely logically, and it's something that mathematicians have been doing for a long time."
Prove it. I'm not sure you can. Of course, that would be because our brain has developed from our senses, and any proof you'd offer would involve a general acceptance as a language being purely logical, which I say would not be possible because we're not purely logical beings. So I suppose I've asked you to do the impossible in proving that, but you might be able to convince me anyways. The rest of that post is moot IMO because you have not proven the above statement. If we take math as representing our world around us, 1*1=1 does not need to be proven logically. "Rather, mathematicians work more predominantly in "First Order Logic," a mathematical language that uses completely unambiguous language to a satisfactory level of precision." And these are the same mathematicians who've been using logic 'for along time' as you say? My knowledge (which is obviously incomplete, but not necessarily wrong), is that 'formal' math in fact started as geometry, which is clearly an offshoot of our visual sense. Also, why is our language of math displayed visually? Yes, I understand that my arguments are far from proof, however, that does not mean they're not valid to consider. |
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First order logic was not developed in Ancient Greece or anything... the whole concept of mathematical logic and consistency is a relatively recent phenomenon. I'm not really up to writing a huge long post about first order logic, but suffice to say that it's precise and unambiguous. And, yes, language in general is not logical, but it is possible to fix all the ambiguities in language, and make it absolutely precise (thus, the creation of F.O.L). Humans may not be logical beings inherently, but we do have the capacity to think logically and unambiguously if we really try. And really trying in that capacity is essential to understanding mathematics. |
Re: Pondering a Proof
I've studied the basics of first order logic.
It seems to be on topic if I asked what makes logic valid in the first place...seems almost like I'm asking why 1*1=1. |
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