How do you know you are not "A.I."?

[Acim]

The problem is that even though it was believed that 0.999... and 1 were the same using informal arguments before the epsilon-delta definition of limits was rigorously formulated, it was BECAUSE of logic and DUE TO logic that the equivalence was proved. It is fairly non-trivial to take the explanations present in the thread and translate them into a formal logical proof, as this is at the heart of analysis. So I have to ask, what do you mean by logical proof?

Short answer: sound reasoning.

With a little elaborating, and sticking to my own wording, I would say it would help (in establishing further understanding) if the reasoning is able to draw on a discipline other than the one that is perceived to be 'only path' for obtaining the 'soundness' of the reasoning. Such that if only math is used, let that be considered at all times as the best path for further understanding, but not the only path. Indeed, I routinely see math mixed in with language/symbols that are not inherently mathematical (as your examples below show), but is making vast assumptions in the non-math language. If those assumptions (or even let's just say language) are going to be employed, then sound reasoning to me ought to be able to explain in other terms, which might not provide for 'best understanding' but do still further understanding.

Hopefully that all makes sense.

Ok, let me give you another restatement of this but now followed by a proof:

Now, how does one logically proof that e.g., 0.999.... converges to 1? First, here's a definition of convergence:

This is not working for me in understanding convergence. This is not the first time I've tried to understand convergence. I acknowledge that convergence plays a (important) role in solving/working with mathematical problems. My choice to understand it thus far has only been to see how it is being utilized in resolving the .999... = 1 problem. Thus far 'limits' and 'convergence' come across to me as trumped up terms being employed to justify what I see as assumptions (or leaps of faith) being made within the argument, that 'of course it equals 1.' Akin to a 'god of the gaps' type rhetoric ploy. Which means it could be that, or it could be that I don't have full understanding of / appreciation for convergence. But routinely, in my attempts to understand convergence, it consistently is explained in words of 'telling' rather than actually explaining. Throw in mathematical symbols while providing the 'explanation' and I'm sure it makes sense to some, but I still see it in how (little) I understand the math and 'explanation' to be making assumptions. So, I wonder... would it be possible to explain convergence and limits (as used in math) without using math, at least at some point of explaining the term. I think a definition might work, but if asking for further understanding, would math then have to be employed to convey that understanding? If yes, I will earnestly make another attempt to understand. But just realize I am doing so with constant reminder to myself to see how it pertains to the .999... problem.

Even this proof is an attempt by the authors (Hubbard & Hubbard from Vector Calculus, Linear Algebra, and Differential Forms, 3rd Ed.) to render the logical proof into more intuitive language. In a course in formal logic, one learns early on how to prove/derive conclusions logically that have no meaning, because logical proofs are not concerned with soundness but validity (i.e., granting some set of premises are true, then necessarily some conclusion follows).

Well, that's interesting. Given how I chose to provide, in my own words, what I see logical proof as meaning. And given that my computer's dictionary defines validity as: being logically or factually sound. Also interesting is the 'granting some set of premises are true' part. That to me, strikes me as significant assumption. Akin to saying, "granted that God does in fact exist, let us now explore some necessary conclusions that follow from that." Moreover, it seems like once the original assumptions are taken as true (or taken for granted), then the resulting conclusions may lead to some really fascinating information or possibly dead ends, but allow an individual to say, "we already know God exists, we are now understanding more about God!"

The whole 'granting of premises' is item I have in various ways wrestled with for a long time. I don't deny that most to all of my assertions, propositions rest on such a rhetorical device. Though it seems like a really huge (monumental) assumption is being made, and then when conclusions are reached, only the validity/soundness of those are addressed or scrutinized. Unless you are not an adherent to that path. Then suddenly everything comes into question, while adherents are busy exploring what follows from conclusions made.

Which is how I see the .999... = 1 problem. We get to assume that is correct so that I can now follow along with convergence and limits to see how it works there and in many other ways. Though since I lack that acceptance, you can rest assured that I will be scrutinizing the heck out of the basic premises of convergence and limits.

 

[Revoltingest]

... would it be possible to explain convergence and limits (as used in math) without using math.....

I recall an old story.....
You travel from Point A to Point B.
After a while, you get halfway to B.
But you still have half the total distance left to travel.
After a shorter while, you've traveled an additional half of halfway to B.
This is an infinite series of traveling half of the remaining distance to B.
But we know that it is possible to travel all the way from A to B.
The infinite series does indeed converge upon the distance from A to B, & they're exactly equal.

 

[Scott C.]

Pretty much... just...... How do you know you are not "A.I."?

I have no reason to believe that I am, so I'll go with that.

 

[Acim]

I recall an old story.....
You travel from Point A to Point B.
After a while, you get halfway to B.
But you still have half the total distance left to travel.
After a shorter while, you you've traveled an additional half of halfway to B.
This is an infinite series of traveling half of the remaining distance to B.
But we know that it is possible to travel all the way from A to B.
The infinite series does indeed converge upon the distance from A to B, & they're exactly equal.

Sounds like a koan.

The part I highlighted seems like a transitional point to greater understanding and wondering if that is worded correctly. If not, perhaps what was intended could impact the overall meaning?

I'm familiar with this 'story' and thought I heard it conveyed differently. I don't get how it relates to convergence and limits, but feel it might be a great way to understand.

For now, I'm going to add to the discussion my computer's dictionary definition of converge to hopefully help with the discussion and (my) further understanding.
Converge = (people or things) come together from different directions so as eventually to meet.

I routinely make a trip that I understand to be 80 miles. Because at times that trip bores me, I look for points that are say 1/3rd the total distance or halfway or 2/3rds, as it makes me think 'not too much longer until I get there.' So, there is a point A (home) and point B (destination) and then points along the way that I have not attempted to accurately measure, but do correspond for me as fractions of the total distance between A and B. Let's say the point I estimate to be 1/4th of the way there is C, half way is D and 3/4ths the way is E.

I feel like I could relate what the story is saying to how I conceive of these distances. But I am reluctant because I do not conceive of the distances during the trip as infinite series, but rather finite. Thus, I feel my trying to reflect back understandings that hopefully lead to me saying, 'ah now I get it' would come across as gobblety gook. But I welcome anyone reading this to use my example (or another) to aid in understanding.

 

[Revoltingest]

Sounds like a koan.
The part I highlighted seems like a transitional point to greater understanding and wondering if that is worded correctly. If not, perhaps what was intended could impact the overall meaning?

I've corrected my post.

I'm familiar with this 'story' and thought I heard it conveyed differently. I don't get how it relates to convergence and limits, but feel it might be a great way to understand.

It relates because traveling a finite distance (call it "D") can be expressed as the sum of an infinite series which converges upon D.
They are exactly equal, as can be observed by measuring D.
This shows that convergence limits can be exactly equal to something as simple as an integer.

For now, I'm going to add to the discussion my computer's dictionary definition of converge to hopefully help with the discussion and (my) further understanding.
Converge = (people or things) come together from different directions so as eventually to meet.

I routinely make a trip that I understand to be 80 miles. Because at times that trip bores me, I look for points that are say 1/3rd the total distance or halfway or 2/3rds, as it makes me think 'not too much longer until I get there.' So, there is a point A (home) and point B (destination) and then points along the way that I have not attempted to accurately measure, but do correspond for me as fractions of the total distance between A and B. Let's say the point I estimate to be 1/4th of the way there is C, half way is D and 3/4ths the way is E.

I feel like I could relate what the story is saying to how I conceive of these distances. But I am reluctant because I do not conceive of the distances during the trip as infinite series, but rather finite. Thus, I feel my trying to reflect back understandings that hopefully lead to me saying, 'ah now I get it' would come across as gobblety gook. But I welcome anyone reading this to use my example (or another) to aid in understanding.

To express a distance as a finite series is also possible, but this is irrelevant.
That it can be expressed as an infinite series is what matters.

 

[fantome profane]

I recall an old story.....
You travel from Point A to Point B.
After a while, you get halfway to B.
But you still have half the total distance left to travel.
After a shorter while, you've traveled an additional half of halfway to B.
This is an infinite series of traveling half of the remaining distance to B.
But we know that it is possible to travel all the way from A to B.
The infinite series does indeed converge upon the distance from A to B, & they're exactly equal.

A very old story.
https://en.wikipedia.org/wiki/Zeno's_paradoxes

 

[Revoltingest]

A very old story.
https://en.wikipedia.org/wiki/Zeno's_paradoxes

Leave it to Zeno to say something better than I would.
I'm glad he's dead!

 

[fantome profane]

Leave it to Zeno to say something better than I would.
I'm glad he's dead!

Actually my favourite version goes like this.

Three men walk into a room, one is a philosopher, one is a mathematician, and one is an engineer. On the far side of the room is a beautiful naked woman beckoning them over. The philosopher and the mathematician sigh deeply knowing that because of Zeno's paradox it is impossible for them to reach the other side of the room. But the engineer strides boldly forward. The other two men are agast, "you fool!" they shout, "don't you understand that you can never get to where she is?". The engineer replies, "Yes of course I know that, I know the math. But I figure I will get close enough".

 

[Revoltingest]

Actually my favourite version goes like this.

Three men walk into a room, one is a philosopher, one is a mathematician, and one is an engineer. On the far side of the room is a beautiful naked woman beckoning them over. The philosopher and the mathematician sigh deeply knowing that because of Zeno's paradox it is impossible for them to reach the other side of the room. But the engineer strides boldly forward. The other two men are agast, "you fool!" they shout, "don't you understand that you can never get to where she is?". The engineer replies, "Yes of course I know that, I know the math. But I figure I will get close enough".

I had no idea there was a joke with the engineer coming out on top!
Usually the engineer is the one who diagnoses the problem with the blade hanging up on the guillotine he's strapped to.

 

[LegionOnomaMoi]

Is your brain single?

Nope. Divided mind with multiple personalities!

 

[LegionOnomaMoi]

Well, that's interesting. Given how I chose to provide, in my own words, what I see logical proof as meaning.

It's not as counterintuitive or strange as one might think at first. Basically, we can think of logic alone as providing us with a necessary condition for truth, but not sufficient. Logic can determine what is valid, but in order to determine if a conclusion is sound, it must be BOTH valid and have true assumptions. So, for example, any attempt to show anything about the physical world we live in requires the assumption (at the very least) that such a world exists.

Proofs are fundamentally mathematical, and therefore need not relate at all to truth, as in the following:
Premise: I am the King of the ReligiousForums
Premise: The king of RF is the ruler of the universe
Conclusion: I am the ruler of the universe

The above derivation is entirely logical. Given the premises, the conclusion necessarily follows. But, of course, the premises are false. In most cases, this isn't so clear. In physics, for example, there is a very clear, mathematical proof underlying the nonlocality of quantum mechanics called Bell's inequality. Physics is probably the science closest to pure mathematics, and this is as close to a proof as one can get in the sciences. However, we find debates in the physics literature on what this shows and why that goes back basically since Bell's original publication. Why? It's a proof, after all, with all the mathematical precision of 1+1=2. The reason is because reality isn't derived from a set of rigorous axioms like formal systems (i.e., those in logic and mathematics). The proof assumes things like realism.

Just about any argument, proof, conclusion, etc., that is logical is either purely syntactic (i.e., meaningless or merely computational), or is based upon assumptions that must be granted. Meaning (semantics) cannot be formalized to obey logical rules of inference, and thus to show some conclusion is sound (true) we must go beyond mere logic.

 

[Ben Dhyan]

I recall an old story.....
You travel from Point A to Point B.
After a while, you get halfway to B.
But you still have half the total distance left to travel.
After a shorter while, you've traveled an additional half of halfway to B.
This is an infinite series of traveling half of the remaining distance to B.
But we know that it is possible to travel all the way from A to B.
The infinite series does indeed converge upon the distance from A to B, & they're exactly equal.

The simplest solution if you want to arrive at Point B with the least hassle is to actually plan your journey to Point C which is double the distance of Point A to Point B...you get to Point B after having only gone halfway...

 

[Revoltingest]

The simplest solution if you want to arrive at Point B with the least hassle is to actually plan your journey to Point C which is double the distance of Point A to Point B...you get to Point B after having only gone halfway...

Story problem for the math impaired.
If a train leaves NYC at 4pm traveling 40mph to St Louis,
& a train leavesk Flagstaff at 5pm traveling 35mph to St Louis,
will either train ever arrive there?

 

[Willamena]

If I were an A.I., I'd be a lot smarter than than I am.

 

[beenherebeforeagain]

Story problem for the math impaired.
If a train leaves NYC at 4pm traveling 40mph to St Louis,
& a train leavesk Flagstaff at 5pm traveling 35mph to St Louis,
will either train ever arrive there?

Possible Answers:
-Yes, but no one aboard will be alive at that point.
-No, because of the paradox cited above.
-Not if I have anything to say about it!

 

[Ouroboros]

Story problem for the math impaired.
If a train leaves NYC at 4pm traveling 40mph to St Louis,
& a train leavesk Flagstaff at 5pm traveling 35mph to St Louis,
will either train ever arrive there?

Awfully slow trains. I don't think either will get there since they both have some machine problem and can't go faster. They'll break down.

Besides, I think train tracks are a sort of a road, and we know that all roads lead to Rome, and St. Louis is not Rome, so no, they can't go there.

 

[Revoltingest]

If I were an A.I., I'd be a lot smarter than than I am.

You already are!
I've long suspected that you're a Turring Test.

 

[Acim]

It's not as counterintuitive or strange as one might think at first. Basically, we can think of logic alone as providing us with a necessary condition for truth, but not sufficient. Logic can determine what is valid, but in order to determine if a conclusion is sound, it must be BOTH valid and have true assumptions. So, for example, any attempt to show anything about the physical world we live in requires the assumption (at the very least) that such a world exists.

I call this faith. Otherwise, it would seem entirely sound that other 'things' that are said to rest on faith could instead be (rational) assumptions.

Proofs are fundamentally mathematical, and therefore need not relate at all to truth, as in the following:
Premise: I am the King of the ReligiousForums
Premise: The king of RF is the ruler of the universe
Conclusion: I am the ruler of the universe

The above derivation is entirely logical. Given the premises, the conclusion necessarily follows. But, of course, the premises are false. In most cases, this isn't so clear. In physics, for example, there is a very clear, mathematical proof underlying the nonlocality of quantum mechanics called Bell's inequality. Physics is probably the science closest to pure mathematics, and this is as close to a proof as one can get in the sciences. However, we find debates in the physics literature on what this shows and why that goes back basically since Bell's original publication. Why? It's a proof, after all, with all the mathematical precision of 1+1=2. The reason is because reality isn't derived from a set of rigorous axioms like formal systems (i.e., those in logic and mathematics). The proof assumes things like realism.

So again, justified by faith in the premise. Not sure why I'm emphasizing this, but also not sure why you are conveying what you are, so just choosing to respond in way that makes sense. And perhaps because I do see .999... equaling 1 as a faith proposition.

Just about any argument, proof, conclusion, etc., that is logical is either purely syntactic (i.e., meaningless or merely computational), or is based upon assumptions that must be granted. Meaning (semantics) cannot be formalized to obey logical rules of inference, and thus to show some conclusion is sound (true) we must go beyond mere logic.

Okay.

 

[Acim]

So again, justified by faith in the premise. Not sure why I'm emphasizing this, but also not sure why you are conveying what you are, so just choosing to respond in way that makes sense. And perhaps because I do see .999... equaling 1 as a faith proposition.

Quoting myself because I do now think I can tie in the .999... side topic with A.I.

Because it is unresolved for me (and pretty sure I'm not alone in this, but do recognize there is a majority view that asserts itself as having this perfectly, and logically worked out), it does seem to pertain to how we set up artificial intelligence. Perhaps not so much in current practice. But in the way we imagine A.I. could go if it were to exceed our understandings. I'm thinking of science fiction stories, i.e. 'Law Of Robotics' or Skynet. Essentially, we have faith in our ability to reason and thus if applied that to A.I. we'd feel comfortable in setting up a program whereby A.I. could not harm humanity. Yet, our sci-fi tales are clearly showing we may be mistaken about that, regardless of how we set up the program. Thus a faith proposition/assumption that is built into 'intelligence' in an attempt to game the system (where humans can't be hurt by such intelligence). And yet, in many to most of our tales, humans don't fare so well in them. Yes, that is partially due to creative licensing, but we certainly have opportunity to tell ourselves how it could work out and be puppies and rainbows for all of humanity when taken to certain 'logical conclusions.' I feel familiar enough with sci-fi to have idea that it could work out well, but am possibly overlooking or not aware of some tales where it explains fairly well how it would work out just fine, or within our ability to maintain control, rather than lead to our demise. Star Trek comes to mind as example of tale where A.I. doesn't lead to our demise (though obviously in that universe, humans aren't only ones creating A.I. and Borg narrative written by Borg writers may not be all hunky dory for rest of universe).

Anyway, I ramble. I do still see .999... as faith proposition. Sorry if that doesn't jibe with anyone reading this. I feel open to considering otherwise, if / when it can be shown to me and stand up to scrutiny, that is really just basic inquiries at this point.

 

[LegionOnomaMoi]

I call this faith. Otherwise, it would seem entirely sound that other 'things' that are said to rest on faith could instead be (rational) assumptions.

It rests on truth. Any argument that something follows from something else (e.g., that if you are a bachelor, you are not married; if you are a billionaire, you must have at least a million dollars; etc.) rests first on whether or not the inference is valid. Logic is what allows us to conclude that something does or doesn't follow from something else. That's validity.
However, logic is more general. I can determine that something follows from something else even if that something else isn't true.
The great thing about mathematics is that the "world" or "universe" we are dealing with is entirely and explicitly defined (it is "closed" or constitutes a "closed discourse universe"). If I say that grass is green, it is either true or false depending upon not only what the words "green" an "grass" mean but the extent to which these terms combine to make a statement which corresponds to external reality. This isn't true of statements like 1+1=2 or that division by 0 isn't defined or that a negative number multiplied by another negative number yields a positive number. In mathematics, everything is either true by definition or because it follows from (i.e., "is necessarily true") BECAUSE of definitions.
That 0.999...=1 is true because it follows logically (i.e., "is necessarily true") given that numbers are uniquely defined on the real number line (actually, it is true even if we don't assume irrational numbers exist; it follows from the properties of the set of all rational numbers/fractions). Logic doesn't cease to work because of infinities (although it sometimes becomes less intuitive). As an example, and one involving the power of proof by assumption (proof by contradiction), consider a statement that deals only with the whole numbers: "there is no greatest whole number."
How can I prove that? Well, I assume it is false. This implies that there exists some number n such that there is no whole number greater than n. But given any whole number, adding 1 to it yields another whole number. This means that n + 1 is a whole number. The assumption yields a contradiction, so it must be true that there is no greatest whole number.
Likewise, if I assume that 0.999... is not equal to 1, then there must exist some other number it is equal to which differs from 1. Logically, this means that the number 0.999... differs from the number 1 by some fixed amount. In your assertions that this difference is 0.000...1, either this difference really is fixed, in which case I can show it isn't the actual difference, or it must be allowed to become infinitely small. If it is allowed to become infinitely small, than the difference between 0.999...must be smaller than any possible number, which means there is no difference.
The premises (the assumptions) are simply that numbers have the properties they do (and need not even be an assumption that ALL real numbers do, merely the set of rational numbers/fractions).

So again, justified by faith in the premise.

If by "faith" you mean truth, sure. Every conditional statement ("if x, then y") involves an assumption/premise, e.g., "if you pour a bucket of water on your head, you will get wet". Another way of saying this is "under the assumption that you actually do pour a bucket of water on your head, you will get wet". It isn't faith that makes this statement true so much as it is you actually satisfying the assumption by pouring water on your head (and that you, water, etc., exists, that water is wet, etc.).