The cat and mouse argument misses the point of the epsilon-delta definition You say there are 2 arguments: 1 - Give me any point before 1 and I can give you an nth sum that is closer to 1 than your point/ 2 - Give me any nth sum and I can give you a point that is closer to 1 than your nth sum term. Lets break this down so we understand what they mean 1 - For all x < 1, there exists n such that x distance (x,1) Where n is a natural number, s_n is the nth partial sum and x is a number (2) implies that s_n has distance not equal to 0 to 1, or that s_n =/= 1, so this is equivalent to 3 - For all n, s_n =/= 1. (Since being able to find a closer number implies inequality and not being equal means theres a point inbetween) Now, I think its a bit clearer that these are not equal, and that (1) is in fact a lot stronger than (3). (3) may be said about almost any number you pick (for example, 2, 3, 4, 0.8, 0, -1005, etc) But (1) can only be said about one number (1), (uniqueness of limit), so clearly that means there is something 'special' about 1 and the sequence 0.9,0.99,0.999, ... And epsilon delta lets us understand more these special numbers. Now, as for whether this is useful, I'd say the proof is in the pudding with our modern world.
@adriengrenier8902 You said: "The cat and mouse argument misses the point of the epsilon-delta definition" I beg to differ. You said: "You say there are 2 arguments: 1 - Give me any point before 1 and I can give you an nth sum that is closer to 1 than your point/ 2 - Give me any nth sum and I can give you a point that is closer to 1 than your nth sum term." Here in 'point 1' I have paraphrased the unclear mathematical terminology into a form that I believe is easier for the lay person to get their head around. Indeed, it is the way that mathematicians often describe the epsilon-delta argument to students and lay persons. Then, using the same clear language, I have shown how this argument can easily be turned on its head. Also note that early on in the video I suggest that 0.999... might be thought of as "notation for values that could be produced from an associated set of instructions (algorithm)". What I was getting at was that it is equivalent to the algorithm for the geometric series with 1st term = 0.9 and common ratio = 0.1. We can think of this as the programming language code or even a set of verbal instructions that describes how to generate the sequence of values 0.9, 0.99, 0.999, and so on. So we don't have to pretend that we can imagine an actual infinity of nines following the decimal point, as we all know that this is impossible. We can easily imagine a finite set of instructions that, when executed, would start to produce this sequence of values. This was the original meaning of a geometric series. In many of Zeno's paradoxes he described processes that we can now map directly to the geometric series 1/2 + 1/4 + 1/8 + ...and so arguably this is what a geometric series is, nothing more than a finite description of a process. Without Simon Steven's publication in 1594 we might have kept this interpretation. Then in our logical reasoning we might start by considering that 0.999... means one of three things: a constant less than 1, a constant equal to 1, or a process description which is a set of instructions rather than a constant. The cat and mouse argument would then discount the 1st two possibilities. You said: "Lets break this down so we understand what they mean 1 - For all x < 1, there exists n such that x distance (x,1) Where n is a natural number, s_n is the nth partial sum and x is a number" In my eyes, this is far, far, far more unclear than my wording. The language of mainstream mathematics claims to be formal and rigorous whereas it is anything but. A computer language is formal and rigorous because it can be executed in the real world. We can create instructions that would, when executed, go into an inescapable loop. But we cannot go around a loop a non-finite amount of times. Also we can create variables of a certain number type and we can specify properties of any created numbers of that type. But we can't create a non-finite amount of a particular number type and we can't assume that 'all' possible values can somehow be thought of as existing at the same time. We have to distinguish between accurate specification and farcical delusional beliefs. And so when you state things like "For all x < 1 ..." you have immediately introduced a contradiction into your logic. You are effectively saying "let's assume that something that can't exist (i.e. an infinite set) actually exists, then it follows that ...". You have accepted the language and arguments of mainstream mathematics and you genuinely believe it is valid because (I presume) you believe that the evidence that supports its validity is so vast and compelling that to question its validity would be unthinkable to you. I know that there is nothing I can say that will change your mind. The best I can hope for is that you will at least respect my right to question your beliefs just I accept your right to question mine. You said: "Now, as for whether this is useful, I'd say the proof is in the pudding with our modern world". When mathematicians create their mathematical objects and formal rules for their axiomatic systems, where do we suppose their inspiration comes from? If they are being influenced by things in the real world then it should come as no surprise to us that some of their mathematics can be found to be useful in the real world. This success of their mathematics in the real world is then claimed to justify having a foundation based on pure fantasy. However I believe the success is down to the way that they have been allowed to design their fantasy axiomatic systems in order to appear to work in the real world. If their abstract systems are really completely detached from all physical reality then we are left to contemplate many wondrous things. Not only is it wondrous where all these concepts have strangely come from, but it is wondrous that they amazingly appear to be so fantastically useful in the real world! It seems that most people love all this wondrous stuff and so are happy to accept it. I am not one of them. I don't accept that we should all be amazed that this fantasy game with symbols has mysteriously and wondrously turned out to be useful in the real world. And I don't accept that it provides any evidence that fantasy-based maths is far better than some other approach, such as a reality-based approach. In order to accept the mainstream position we have to believe there is some mysterious property of mathematics that is strangely not connected with physical reality but that mysteriously turns out to be useful in the real world. Mathematicians do not appear to be aware of how absurd this claim sounds to people like me. To be absolutely clear, it sounds like they are attributing magical powers to mathematics!
Say we have the series 9/10 + 9/100 + 9/1000 + ... then the nth sum is the sum of all the terms up to the nth term. So the 1st sum is 0.9, the second sum is 0.99 and so on. The value 0.9123 is between the 1st sum and the 2nd sum. Mainstream mathematicians often call numbers 'points' because they imagine that they can 'exist' on an infinitely thin number line.
The cat and mouse argument misses the point of the epsilon-delta definition
You say there are 2 arguments:
1 - Give me any point before 1 and I can give you an nth sum that is closer to 1 than your point/
2 - Give me any nth sum and I can give you a point that is closer to 1 than your nth sum term.
Lets break this down so we understand what they mean
1 - For all x < 1, there exists n such that x distance (x,1)
Where n is a natural number, s_n is the nth partial sum and x is a number
(2) implies that s_n has distance not equal to 0 to 1, or that s_n =/= 1, so this is equivalent to
3 - For all n, s_n =/= 1.
(Since being able to find a closer number implies inequality and not being equal means theres a point inbetween)
Now, I think its a bit clearer that these are not equal, and that (1) is in fact a lot stronger than (3). (3) may be said about almost any number you pick (for example, 2, 3, 4, 0.8, 0, -1005, etc) But (1) can only be said about one number (1), (uniqueness of limit), so clearly that means there is something 'special' about 1 and the sequence 0.9,0.99,0.999, ... And epsilon delta lets us understand more these special numbers.
Now, as for whether this is useful, I'd say the proof is in the pudding with our modern world.
@adriengrenier8902 You said:
"The cat and mouse argument misses the point of the epsilon-delta definition"
I beg to differ.
You said:
"You say there are 2 arguments:
1 - Give me any point before 1 and I can give you an nth sum that is closer to 1 than your point/
2 - Give me any nth sum and I can give you a point that is closer to 1 than your nth sum term."
Here in 'point 1' I have paraphrased the unclear mathematical terminology into a form that I believe is easier for the lay person to get their head around. Indeed, it is the way that mathematicians often describe the epsilon-delta argument to students and lay persons. Then, using the same clear language, I have shown how this argument can easily be turned on its head.
Also note that early on in the video I suggest that 0.999... might be thought of as "notation for values that could be produced from an associated set of instructions (algorithm)". What I was getting at was that it is equivalent to the algorithm for the geometric series with 1st term = 0.9 and common ratio = 0.1. We can think of this as the programming language code or even a set of verbal instructions that describes how to generate the sequence of values 0.9, 0.99, 0.999, and so on.
So we don't have to pretend that we can imagine an actual infinity of nines following the decimal point, as we all know that this is impossible. We can easily imagine a finite set of instructions that, when executed, would start to produce this sequence of values. This was the original meaning of a geometric series. In many of Zeno's paradoxes he described processes that we can now map directly to the geometric series 1/2 + 1/4 + 1/8 + ...and so arguably this is what a geometric series is, nothing more than a finite description of a process.
Without Simon Steven's publication in 1594 we might have kept this interpretation. Then in our logical reasoning we might start by considering that 0.999... means one of three things: a constant less than 1, a constant equal to 1, or a process description which is a set of instructions rather than a constant. The cat and mouse argument would then discount the 1st two possibilities.
You said:
"Lets break this down so we understand what they mean
1 - For all x < 1, there exists n such that x distance (x,1)
Where n is a natural number, s_n is the nth partial sum and x is a number"
In my eyes, this is far, far, far more unclear than my wording. The language of mainstream mathematics claims to be formal and rigorous whereas it is anything but. A computer language is formal and rigorous because it can be executed in the real world.
We can create instructions that would, when executed, go into an inescapable loop. But we cannot go around a loop a non-finite amount of times. Also we can create variables of a certain number type and we can specify properties of any created numbers of that type. But we can't create a non-finite amount of a particular number type and we can't assume that 'all' possible values can somehow be thought of as existing at the same time. We have to distinguish between accurate specification and farcical delusional beliefs.
And so when you state things like "For all x < 1 ..." you have immediately introduced a contradiction into your logic. You are effectively saying "let's assume that something that can't exist (i.e. an infinite set) actually exists, then it follows that ...".
You have accepted the language and arguments of mainstream mathematics and you genuinely believe it is valid because (I presume) you believe that the evidence that supports its validity is so vast and compelling that to question its validity would be unthinkable to you. I know that there is nothing I can say that will change your mind. The best I can hope for is that you will at least respect my right to question your beliefs just I accept your right to question mine.
You said:
"Now, as for whether this is useful, I'd say the proof is in the pudding with our modern world".
When mathematicians create their mathematical objects and formal rules for their axiomatic systems, where do we suppose their inspiration comes from? If they are being influenced by things in the real world then it should come as no surprise to us that some of their mathematics can be found to be useful in the real world. This success of their mathematics in the real world is then claimed to justify having a foundation based on pure fantasy.
However I believe the success is down to the way that they have been allowed to design their fantasy axiomatic systems in order to appear to work in the real world. If their abstract systems are really completely detached from all physical reality then we are left to contemplate many wondrous things. Not only is it wondrous where all these concepts have strangely come from, but it is wondrous that they amazingly appear to be so fantastically useful in the real world! It seems that most people love all this wondrous stuff and so are happy to accept it. I am not one of them.
I don't accept that we should all be amazed that this fantasy game with symbols has mysteriously and wondrously turned out to be useful in the real world. And I don't accept that it provides any evidence that fantasy-based maths is far better than some other approach, such as a reality-based approach.
In order to accept the mainstream position we have to believe there is some mysterious property of mathematics that is strangely not connected with physical reality but that mysteriously turns out to be useful in the real world. Mathematicians do not appear to be aware of how absurd this claim sounds to people like me. To be absolutely clear, it sounds like they are attributing magical powers to mathematics!
I don't get it, what's the difference between a point and an nth sum?
Say we have the series 9/10 + 9/100 + 9/1000 + ... then the nth sum is the sum of all the terms up to the nth term. So the 1st sum is 0.9, the second sum is 0.99 and so on. The value 0.9123 is between the 1st sum and the 2nd sum. Mainstream mathematicians often call numbers 'points' because they imagine that they can 'exist' on an infinitely thin number line.
Love your channel bro 👍
Glad to hear it
Excellent explanation of the issues with mathematics when applied to the real world. Thank you.
Glad you enjoyed it!