ducknumerouno
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- TL;DR Summary
- Pi = 4?
I know it theoretically never touches the circle, but does the circle ever really become a circle?
This depends on the particular occasion. Infinities in calculus are replaced by the epsilontic and topological principles, infinities in number theory are based on induction, which in turn is based on Peano's axioms.ducknumerouno said:If infinity isn't a number and behaves differently from regular numbers, how can we really understand and use it in things like limits?
Yes, because we have methods to cope with these notations and do not really use "infinity".ducknumerouno said:Can we even trust that it is useable in math then?
You probably heard that there is a difference between countable many infinite numbers such as the integers, and uncountable many infinite numbers such as the real numbers. That makes a provable difference.ducknumerouno said:Because I heard somewhere that some infinityies are bigger than others ( I don't know where I heard that some scientist proved it)
As mentioned, it depends on the context. I cannot explain ##\infty ## without knowing that context.ducknumerouno said:I get that infinity is handled differently in different areas of math, but it still feels like we’re just ignoring how weird it is. Calling infinity a “shorthand” doesn’t really solve the problem—it just makes it easier to deal with, but doesn’t explain it fully.
So my edit to the previous post.ducknumerouno said:rue, infinity definitely depends on the context, but that just shows how confusing it can be to some people . If it can change meaning based on the situation, maybe it’s more of an issue than we know?
Well, to you, clearly, but not to most of us. Study some math and you'll get over it.ducknumerouno said:... that just shows how confusing it can be to some people
No. MANY words in the English language have different meanings depending on context. Does that make them philosophical things rather than real words?ducknumerouno said:But if infinity is always context-dependent and never truly defined, doesn’t that make it more of a philosophical idea than a mathematical thing?
I understand your point, but just because infinity is useful in math doesn’t mean it’s fully understood or without its philosophical issues. It’s a concept that helps explain things, but the fact that it’s treated as something both undefined and context-dependent makes it weird. Learning about it doesn’t erase the questions, understanding something means recognizing that it’s not as clear as it seems?phinds said:Look, @ducknumerouno, it should be clear to you from the responses in this thread that infinity is a mathematical concept that is quite useful. If you wish to view it as something else, feel free to do so, but do not expect the rest of the world to agree with you. Learn some math and you'll get over it.
You are complaining about something you don't understand. Instead of doing that, why not spend some time LEARNING about it.
You only believe it's not clear because you refuse to actually LEARN about instead of just complaining.ducknumerouno said:Learning about it doesn’t erase the questions, understanding something means recognizing that it’s not as clear as it seems?
Seriously ??? What do you think DOES provide understanding? Voodoo? Magic?ducknumerouno said:There is no problem with learning, it just provides little understanding
##2\cdot 2=4## and ##2\cdot 2 =1## are both true statements but, obviously, the twos represent something different in each statement.ducknumerouno said:I understand your point, but just because infinity is useful in math doesn’t mean it’s fully understood or without its philosophical issues. It’s a concept that helps explain things, but the fact that it’s treated as something both undefined and context-dependent makes it weird. Learning about it doesn’t erase the questions, understanding something means recognizing that it’s not as clear as it seems?
True.ducknumerouno said:Knowledge is a paradox. The more you learn, the more you realize how much you don't know.
It is true in the world of remainders by division by three. If we multiply two numbers, say ##5## and ##14## that both have the remainder ##2## then ##5\cdot 14=70## has the remainder ##2\cdot 2=1.##ducknumerouno said:How exactly is 2 * 2 equal to one a true statement,...
A deeper understanding starts with the insight that symbols mean different things in different contexts. There is no one and only infinity in mathematics. We use it as a symbol, and the context defines what it means. I could list quite a few, but that won't bring you closer to an understanding of infinity. It is precisely defined whenever we use it without having to refer to something infinite. That avoids the philosophical dimension in mathematics at the cost that we mean different things in different contexts if we use the symbol ##\infty. ##ducknumerouno said:... and learning can erase questions but you are not deeply understanding ...
I did not use a comparison; I only mentioned two (among even more) possibilities.ducknumerouno said:... instead of saying "light weigh between light that brigtens my eyes" which proves you learned at some point that there was a difference between the two proves that you are thinking very simply mindedly why not say, "I can distinguish the meanings of the word light from the other meaning with the same spelling" instead of directly comparing them? also you could be asking yourself what are the other meanings and how can I compare this better?
It should have shown that there is no single concept of infinity in mathematics. I even doubt that there is a single concept of infinity in philosophy. Hence, if you search for a deeper understanding, you have to either learn all the different uses and gain a deeper understanding in each case, or study philosophy.ducknumerouno said:I appreciate the comparison but I don't get how this can relate to understanding rather than knowing the concept of infinity?
Just because you can't operate on infinity with arithmetic as if it were a number does not mean you can't learn how we do work with it.ducknumerouno said:I don't get how this can relate to understanding rather than knowing the concept of infinity?
You have a sequence of shapes, ##S_n## say, which converge to some shape, ##S##. In this case ##S## is a circle.ducknumerouno said:TL;DR Summary: Pi = 4?
I know it theoretically never touches the circle, but does the circle ever really become a circle?
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Well, the point of the demo is that, eventally, they become infinitely small.gmax137 said:I would not say the square with knocked off corners "converges" to the circle. The steps might get too small to "see,"
Huh? You lost me there.fresh_42 said:Infinity itself is no mathematical concept.
So what? They're still orthogonal steps, they don't converge to the circle. You're never following a hypotenuse.DaveC426913 said:Well, the point of the demo is that, eventally, they become infinitely small.
The concept of convergence requires some way of measuring the difference between two shapes. You could measure the total area between them, for example. In that sense the shapes converge to the circle.gmax137 said:So what? They're still orthogonal steps, they don't converge to the circle. You're never following a hypotenuse.
You cannot speak about infinity itself in mathematics, i.e., without mentioning the domain. Infinity stripped of any context is not defined in mathematics. In this sense, mathematics cannot contribute any insights to the philosophical question.phinds said:Huh? You lost me there.
Ah. Now I'm back with you. Thanks.fresh_42 said:You cannot speak about infinity itself in mathematics, i.e., without mentioning the domain.
In fact, you can use this method to calculate pi, but you need to consider the area and not the perimeter.ducknumerouno said:TL;DR Summary: Pi = 4?
I know it theoretically never touches the circle, but does the circle ever really become a circle?
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Yes!PeroK said:The concept of convergence requires some way of measuring the difference between two shapes.
You could measure the total area between them, for example. In that sense the shapes converge to the circle.
Not at all. It's simply a sequence of functions converging to a limit function. Those square-wave functions meaningfully converge to a circle. This is also the basis of Fourier analysis, where an infinite series of functions converges to the required limit function.gmax137 said:I think saying, "converge to the circle" by itself, is a vacuous statement.
PS I should have said it doesn't necessarily work with perimeter lengths. If we use regular polygons with increasing number of sides, for example, outside and inside the circle, then we get an estimate of pi. I think that was the original method of estimating pi.PeroK said:It doesn't work with perimeter lengths.
That's true. I mean, the initial square is composed of the exact same infinite number of line segments. All the problem does is mix them up.Dullard said:The initial problem strikes me as sleight-of-hand. Doubling the number of 'sides' while halving their individual lengths results in no change in the total length.
Which is getting close to pi.BWV said:By the same token, if the OP's perimeter method works for circles should it not also work for any polygon inscribed in the square? (but it obviously does not, for example a dodecahedron would have a perimeter of 3.106)
( Dodecagon )BWV said:By the same token, if the OP's perimeter method works for circles should it not also work for any polygon inscribed in the square? (but it obviously does not, for example a dodecahedron would have a perimeter of 3.106)
The word “infinity” is used to name several different things, but that’s a problem with the language not the concepts being named; these are precisely and unambiguously defined mathematical concepts.ducknumerouno said:But if infinity is always context-dependent and never truly defined, doesn’t that make it more of a philosophical idea than a mathematical thing?
Context is everything, as they say ...Nugatory said:it would be nice if the English language didn’t use the same word “suit” to describe the thing I wear to my friend’s wedding, the thing lawyers file in courts, one of the four categories of playing cards