Pondering a Proof

[QED Stepfiles]

Originally Posted by devonin

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]

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]

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]

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]

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]

Originally Posted by Cavernio

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]

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]

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

[Reach]

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]

"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.

[QED Stepfiles]

Originally Posted by Reach

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 >_>

Yes I pretty much think that too, but this is really for the sake of discussion. For the purposes of keeping sane it's not really something we should worry about every time we solve an equation, but it's still interesting enough to think about every once in a while.

[kmay]

i said how the math works... what don't u get?

[QED Stepfiles]

Originally Posted by kmay

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

i said how the math works... what don't u get?

Define division. What does it mean to divide 1/1? Is the operation well defined? Is there an inverse of "1" in the field? Is the inverse unique? What does "*" mean? Is "*" a function (Z x Z --> Z), or what? What are the integers? Some texts define the integers as the intersection of all inductive subsets the reals, but what's your definition? According to your definition, is it viable to say things like "1/1"? And is "*" well defined? And what does "=" mean anyways? Does 1/1=1 mean that 1/1 is congruent to 1 (i.e. in the same equivalency class), or that we may represent the same mathematical object "1" as both "1" and "1/1," or what?

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.

[kmay]

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. >.>

[Cavernio]

"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.

[QED Stepfiles]

Originally Posted by Cavernio

"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.

Actually... the whole formal math starting from geometry thing may be accurate to some degree, but today mathematics is definitely not built up on geometry. Geometry is built up off of logic. Actually... euclidean geometry is a bit dated... there are a hell of a lot of assumptions in euclidean geometry (for example, why do we have to use the standard metric -i.e. measure distance and length the standard way- for the space?). Geometry is a result of logic.

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.

[Cavernio]

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.