The art of mathematical argument

C

Cypress

Well-known member
Mathematical proofs are the gold standard of knowledge.

Once a mathematical statement has been proved with a rigorous argument, it counts as true throughout the universe and for all time.

The experience is the closest you can get to glimpsing the abstract order behind all things.


Notable mathematical proofs

The square root of 2: Can the square root of 2 be expressed as a rational number—that is, as a fraction of two integers? The discovery by the ancient Greeks that square root of two is an irrational number was considered so dangerous, that the geometer Hippasus was murdered to prevent the knowledge from being disclosed.

Geometric series: Is the repeating decimal 0.99999… less than 1? Or does it equal 1? The proof that 0.9999... is actually equal to 1 surprises many people and provides a launching point into the analysis of infinite geometric series in calculus.

Countably infinite sets: An eye-opening proof shows the counter intuitive notion that the set of all rational numbers is the same size as the set of all positive integers, even though there are infinitely many rational numbers between two consecutive integers.



Insipration and credit: Professor Bruce Edwards, University of Florida
 
Cypress said:
Mathematical proofs are the gold standard of knowledge.

Once a mathematical statement has been proved with a rigorous argument, it counts as true throughout the universe and for all time.

The experience is the closest you can get to glimpsing the abstract order behind all things.


Notable mathematical proofs

The square root of 2: Can the square root of 2 be expressed as a rational number—that is, as a fraction of two integers? The discovery by the ancient Greeks that square root of two is an irrational number was considered so dangerous, that the geometer Hippasus was murdered to prevent the knowledge from being disclosed.

Geometric series: Is the repeating decimal 0.99999… less than 1? Or does it equal 1? The proof that 0.9999... is actually equal to 1 surprises many people and provides a launching point into the analysis of infinite geometric series in calculus.

Countably infinite sets: An eye-opening proof shows the counter intuitive notion that the set of all rational numbers is the same size as the set of all positive integers, even though there are infinitely many rational numbers between two consecutive integers.



Insipration and credit: Professor Bruce Edwards, University of Florida
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69 is devine?
 
The mathematical ratios of harmonics are pretty strange.

We can intuitively sense the distinction between a major and a minor chord, even without musical training, and it can even evoke an emotional reaction.

I do not know if harmonics are perceived by any other animals in that way.
 
BidenPresident said:
Math is just math. It's not magic or the order of the universe.
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Where would you get magical from anything I wrote?

I do not believe in magic, but I believe the cosmos is essentially mathematical in it's nature.

The beauty of math is it's pure logic. All our other attempts at philosophizing and inquiry are prisoners of language and linguistic convention.
 
Cypress said:
The mathematical ratios of harmonics are pretty strange.

We can intuitively sense the distinction between a major and a minor chord, even without musical training, and it can even evoke an emotional reaction.

I do not know if harmonics are perceived by any other animals in that way.
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Now that is interesting. I imagine that for many creatures, sound = communications = emotions = action. Whale song, bird song, echolocation, etc. are all communication. With the right instrumentation, our human ears can detect the harmonics in them.
 
ThatOwlWoman said:
Now that is interesting. I imagine that for many creatures, sound = communications = emotions = action. Whale song, bird song, echolocation, etc. are all communication. With the right instrumentation, our human ears can detect the harmonics in them.
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I forgot about Whale songs!

I wish my father were around, because he was a professional musician, and it is a mystery to me why certain harmonics sound pleasant to us, and others downright discordant. We seem to be basically tuned to the mathematics of harmonics
 
Cypress said:
Where would you get magical from anything I wrote?

I do not believe in magic, but I believe the cosmos is essentially mathematical in it's nature.

The beauty of math is it's pure logic. All our other attempts at philosophizing and inquiry are prisoners of language and linguistic convention.
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Math has syntax. It's all just code or symbols. Math is no different from language.
 
BidenPresident said:
Math has syntax. It's all just code or symbols. Math is no different from language.
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If math were strictly the equivalent of spoken language, one could fully and comprehensively explain quantum mechanics just by using vernacular English.


Languages should easily be able to be translated and interpreted into other languages, as professional interpreters prove every day.
 
Cypress said:
If math were strictly the equivalent of spoken language, one could fully and comprehensively explain quantum mechanics just by using vernacular English.


Languages should easily be able to be translated and interpreted into other languages, as professional interpreters prove every day.
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Math and the English language are codes. Just different symbol systems.
 
BidenPresident said:
Math and the English language are codes. Just different symbol systems.
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Russian, Tagalog, and Swahili can easily be translated into English.

I have never met anyone who could really translate the meaning and significance of the quantum wave function into English.

I think math is sort of like a language, but it is something far more than a language. The value of Pi, the value of e, and the Fibonacci sequence are intrinsic properties of reality.
 
Cypress said:
Russian, Tagalog, and Swahili can easily be translated into English.

I have never met anyone who could really translate the meaning and significance of the quantum wave function into English.

I think math is sort of like a language, but it is something far more than a language. The value of Pi, the value of e, and the Fibonacci sequence are inherent properties of reality.
Click to expand...

Language is a code.
 
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