Pondering a Proof

[QED Stepfiles]

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.

Again, that's a bit of an oversimplification. How do we know when math has become "broken"? As far as we know, it may be broken already, even if we have not found a way to prove it is so. What you're suggesting is really a proof by lack of counterexample... which is not a very good way to rigorously define things in mathematics. The fact of the matter is that rigorous proofs of whether or not mathematics as a system is consistent are few and far between... at least as far as I can tell (I'm not a logician though, so don't quote me on that).

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.

[Patashu]

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

[QED Stepfiles]

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

Actually, you'd be pretty hard pressed to define a coherent mathematical system based on the axiom 1*1=2. There's a reason why 1*1=1... because given the most natural set of axioms, it is important to define an identity element.

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...

[devonin]

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.

[MrRubix]

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.

[QED Stepfiles]

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.

This isn't really a concern of linguistics. We are not talking about "one" and "group" in a physical sense - indeed, the generalization of mathematics to something more tangible is a bit irrelevant here. Instead, we are treating "one" and "group" as mathematical concepts, and as purely ideal mathematical objects. You're making another assumption in that 1*1=1 is a simple expression that relates to some physical set of objects. There is still a major concern of whether or not such a leap is even allowed (and we have no reason to believe that it is not, but it still is a leap that should not just be treated so callously as to say it's "obvious").

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.

[devonin]

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.

[QED Stepfiles]

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.

[Cavernio]

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.

[Reach]

Re: Pondering a Proof

Well, math is certainly a language.

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

Not *necessarily*, from the simple fact that just because a formal system cannot formally prove something does not mean that it is untrue. This is just a limitation of any formally defined system.

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.

[QED Stepfiles]

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.

Mathematics doesn't appear to me to have a primary reason of explaining the world around us. Hell, most of the stuff I've done in algebra this year to me seems to have no physical significance whatsoever. Showing 1*1=1 may have some physical meaning attached, but proving the Sylow Theorems... not as much. Just because something is mathematically interesting does not really mean that it necessarily have any tangible analogue in the real world. If we manage to find one, kudos to us, but it's doubtful that the mathematician who first came up with the theorem had any physical application in mind.

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.

[dooey100]

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.

[kmay]

Re: Pondering a Proof

well 1*1=1 so divide over the 1 it would be 1=1/1 umm its pretty simple...

[Reach]

Re: Pondering a Proof

Ultimately, to me it seems that we will never be able to prove everything in mathematics.

Well, you're right, as Kurt Godel has already shown through the famous incompleteness theorem that mathematics (and all formal systems) is fundamentally and will always be incomplete.

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 >_>

[MrRubix]

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.