You Can Know Things Are True But Can't Prove Them True

dividavi
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You Can Know Things Are True But Can't Prove Them True Apr 1, 2018 1:27:57 GMT

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Post by dividavi on Apr 1, 2018 1:27:57 GMT

To start off, you might want to know what is meant by irrational numbers and transcendental numbers. An irrational number, such as the square root of 2, is a number that can not be represented as a fraction X/Y where X and Y are integers. A transcendental number is always irrational but has the additional property that it can not be a solution for a polynomial equation where all identified numbers are integers. The square toot of 2 is irrational bur nit transcendental.

See here for more : en.wikipedia.org/wiki/Transcendental_number

There are those numbers listed on that wikipedia page where there are no proofs as to whether a particular real number (no imaginary numbers included) is transcendental or not. Here are some more I invented:

A=0.1223334444555556666667777777888888889999999991010101010101010101010 (one 1, two 2's, seven 7's, ten 10's, etc)
B=0.2357111317192329313741434753596167717379838997101103107109113127131137139149151 (successive prime numbers)

Take A and B, add them together, multiply one by the other, use them for exponentials and you have more transcendental numbers. There's no proof that I know of for those numbers being transcendental and it may be there never will be or even can be. Even if somebody comes up with a proof I can always create odd sequences that represent transcendental numbers.

No proof is required to know those numbers are transcendental. It's already obvious that they are.

Eva Yojimbo
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You Can Know Things Are True But Can't Prove Them True Apr 1, 2018 3:16:50 GMT

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Post by Eva Yojimbo on Apr 1, 2018 3:16:50 GMT

It will take someone more mathematically inclined to address your specific examples, but your thread title sounds similar to Godel's Incompleteness Theorems, since the first states: "For any such formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system."

However, I question whether knowing "things" about numbers is all that similar to knowing things about objective reality. I'm not a mathematical realist, so I think that numbers are merely a part of our mental "maps" rather than part of reality's "territory;" they're simply a very effective way in which we model the territory. Depending on the rules we set up for numbers, we end up with categories like "transcendental numbers" that might merely descriptions of elements within that system. I don't know if we can meaningfully say that this counts as "knowing things" in the same way that we might "know" that, say, we're typing a reply on a keyboard.

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phludowin
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You Can Know Things Are True But Can't Prove Them True Apr 3, 2018 18:24:50 GMT

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Post by phludowin on Apr 3, 2018 18:24:50 GMT

pi is transcendental. 1-pi is transcendental as well.
The sum is not. This is explained in the Wikipedia article. The numbers are not independent.

Maybe the definition of "independent" in that sense is that when you perform algebraic operations on two transcendent numbers, the result will be transcendent as well. I don't know; it's been a while since I finished my Maths studies.
I remember the definition of a transcendental number: It can not be constructed with compass and ruler.

e^2pi*i = 1 in complex numbers. But you excluded complex numbers.

So I guess you are right. Some numbers can't be proven to be transcendent, yet you know they are.

Common sense is another word for prejudice.