Zino said that Achilles will never catch up with the turtle, because when he chases half, there is still half, and when he catches half, there is still half. If he continues to talk recursively, then Achilles will never catch up with the turtle. So, can this problem be solved after calculus is generated? The distance Achilles chased the turtle = 1/2+1 /4+1/8+... ... It is a converging series f, lim(f)=1, so Achilles caught up with the turtle with only one step (that is, 1s). Is it solved?
looks perfect, slower, there is a loophole, that is, for infinite sum sequences, we think here f=lim(f). Why are they equal? Because the series converges -f=lim(f). So why can the series converge f=lim(f) be deduced? Because f is infinitely close to lim(f). Why is f infinitely close to lim(f)? Because the series converges. You can see that the above proof contains a vicious cycle. To prove that f=lim(f), the strictness of this equal sign is equivalent to proving that f(x)=2^(-x) is within the range of x0 and has an intersection with the x axis! And this is impossible to prove. There is always a difference between
f and lim(f). No matter how big N is, f and lim(f) are not equal. When N is infinity, there is an infinitely small black hole. However, the diameter of this black hole is 0, and consciousness is not captured by the black hole, but undergoes a quantum jump of time 0 to reach the other end of the black hole--I saw Achilles surpassing the turtle in a flash.
Can I prove that f(x)=2^(-x) and the x-axis have an intersection point at infinity? Imagine this diagram, draw an infinite x-axis in your mind, which intersects the x-axis. So what about the curve on the right side of the intersection point? Can y be taken to a negative number based on the median theorem? Or the reason why infinity is infinity is because it is the end, and there is no further right. So if there is an end, is it still infinite? N is infinity, but is N+1 still greater than N? Does infinity exist? We seem to solve the Zeno paradox, but at the cost of introducing a new paradox - just like we are trying to reach the end of the Mobius belt! ! ! ! ! ! ! So much so that this proof process is always cycling, and the process of proof has no end - it seems that we are climbing a ladder with no end, as if the truth is within reach in front of us, but we find that no matter how many layers we climb along the ladder, the distance has not decreased. If this proof process exists, it will be infinite! Just like the existence of irrational numbers and transcendent numbers, it is impossible to calculate with finite algebraic expressions.
Pythagoras said that if the universe stops letting him measure the state of each atom, he can predict everything in the future. He made three mistakes. One is that during the measurement process, the influence of the measuring tool and the measurement tool itself cannot be measured. Just as there is no measuring tool that can measure itself, this measurement is incomplete; the second error is that if time stops, the movement stops, and the phenomena and characteristics stop - the light stops propagation, nothing can be seen, and the measurement itself cannot be carried out. The third error is that even the shortest time contains infinite information. Think about how many atoms there are in the universe, so if you need to express it with finite rules, the measurement time must be very short-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Therefore, the three constraints of time, space, matter, and information determine that a complete measurement is impossible, and a complete regularity understanding will not be possible. All laws are just short-sighted and paranoid guesses.
Zeno is not an idealist, but an ancestor of agnosticism. Let’s demonstrate Bi’s theory. We see that there is no absolute truth, the future is unpredictable, all laws are unknown, and the unpredictable theory of quantum mechanics reveals its smile.
[Story of Singularity 1]
Achilles is practicing weightlifting and encounters a turtle.
Turtle asked: Can you lift the 50 kilogram sandbag?
A: Of course there is no problem for a person as strong as me.
Wu: Can you lift 500 kilograms?
A: I am not Optimus Prime, 500 kilograms are not good.
Wu: What about 100 kilograms?
A: It is probably possible to lift
Wu: Where is the 200 kilograms?
A: It depends on my condition. Maybe I can lift it when I am in a good mood, which is a 50% chance.
Wu: Where is the 210 kg?
A: Well, maybe 40% probability
Wu: What about 250 kg?
A: 1% probability
Wu: What about 260 kg?
A: 0.01% probability, haha, if you add weight to me, the probability of I can lift it gradually decreases, approaching 0.
U: When will the probability of approaching 0 become 0, just like the probability of 500 kg is the absolute value of 0?
Ah: (Speechless)...
Achilles encountered a singular point problem: he knew that there was a singular point, connecting = 0 and 0 but very close to the two concepts of 0. But where is this point, it cannot be proved at all: because any proof is absurd. But Achilles clearly felt that this point exists, but he just couldn't say it out and could only comfort himself. "There is a point where convergence to its limit equals the object to be studied." This sentence is irreproveable, but it is clearly valid. Therefore, the methods of scalar and vector have failed, and Achilles can only resort to set theory and formal language: there is a quantifier that can be expressed, and there must be such a point. But I don't need to care about where this point is.
Look at mathematical analysis, the problems of limits, various existence theorems, median theorems, and convergence problems, aren't they all the sandbag 0 points raised by Achilles? If a concept about numbers, such as x*x=2, what is x, the answer itself cannot be expressed by numbers, but it is the only exact answer, we invent a form of symbol (root number) to make up for the shortcomings of the expressive ability of numbers themselves. Numbers are incomplete and incomplete, and the so-called "concept" cannot even express itself in many cases.
[Story of Singularity 2]
also likes to study problems, and loves to be straight. At one time, he went out to a restaurant with a friend from Xi'an. He ordered dishes, mutton soaking buns + mutton scorpion. You should know that for a southerner, eating this for the first time is no less than being punished - I watched him eat with relish, and he looked at me with sorrow, so a conversation began:
"How do you think this thing tastes?"
"Taste is OK", out of politeness, it cannot hit your friends, "Isn't it too used to it yet"
"Just just say it's not delicious?"
"Haha, why do you think it's delicious?" I asked back.
"I have thought it tastes delicious since I was a child"
"Then why didn't I think it taste so delicious?" I continued to question.
"Maybe you feel fear in your heart"
"If it's the first time you've eaten it, can you guarantee it's delicious?"
"Well, this is hard to say. Maybe you'll find it delicious after eating it a few more times?"
"Then, you say that this mutton soup bun is delicious, not because it's delicious, but just because you're used to its taste."
"Maybe, maybe habit is a kind of beauty"
"So that, if we're used to something, we can accept it, and then we think that this thing is good---just like if we think fat is beautiful since we were young, then there are many people who will chase "
" that seems to make sense. But I think a good thing, a beautiful thing, should be something we expect.For example, if you want sweetness, you say it tastes delicious. "
"A man imagines holding a beautiful woman. Since he wants to hug it firmly, the beauty's waist cannot be thick. So in reality, the standard for beauty has a thin waist"
"Well, it seems that this is the case. In other words, it is not because a beauty has a thin waist, we think that thin waist is beautiful, but we hope for something like a thin waist from the heart, and then we put it on, score it, and then judge who is a beauty. "
"So, the standard for judgment does not come from the outside world, but only from our hearts"
"Just like the dots, lines and surfaces in mathematics, go to reality and grab one and take a look? In fact, it is just a concept in our hearts. "
"Does there any standard for beauties?"
"Imagine what you like: smooth feeling, fresh tone, soft material, then the standard for beauty must be white teeth, red lips, ice skin and jade bones. This is the same as when you bought a piece of jade. Beauty is the hope in the heart, not the objective standard "
". Okay, let's go back to the issue of eating. Why do you think this is delicious or not delicious? "
"It's delicious because this taste is something I've been looking forward to before eating. I know what taste I will get, and the process of eating is as I wish; and for things I haven't eaten, its taste is far from our expectations, and it will naturally cause our disgust. "
"In other words, the so-called beauty is just that our own wishes are recognized and realized. "
"Good, just like listening to music, why canon is so nice? Because when I heard the first few notes, I already knew what the last few notes were. If I can't predict the following notes, then there is almost no doubt that what I hear is Gaussian white noise "
"Well, beauty is the realization of self, concept is the realization of self, and even science is the realization of self,"
"Just as Plato said, knowledge also comes from meditation"
"At least I don't think it can be refuted"
"Does the imaginary unit i exist? How to prove it?"
"It only exists in our psychology, and even the real number 1 only exists in our psychology, and cannot be proved."
truth seems to be a kind of harmony of thought, just like Canon, not the height of the level, but a kind of entanglement of the self. Those concepts of mathematics, plural numbers, analysis, limits, infinity, expansion, approximation, orthogonal, duality, set, space, group, deduction, not only reflect the beauty of nature and the beauty of the universe, but not the beauty of mathematics itself. For example, it is a kind of beauty consciousness and desire from our hearts, which has already been subjectively hoped for "simplicity and beauty". Beauty does not come from the object world, beauty only comes from our hearts. We construct truth and reveal the universe according to this desire. And that object world seems to be just a pair of meaningless connections between probability changes and random processes, just because we open our eyes, meaning is generated.
This article is reproduced from the official account "Love Mathematics Home"
