Mathematics - 2016-12-13 (page 5 of 5)

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Secret

23:07

Last night dream strongly reminds of a mix of the hydra game, cayley tables and social networks. Basically there's a process of unwrapping a statement of uncertain truth value.

Suppose there's a statement saying "Y is supported by the research of X". Then for every such statement, a table can be generated as follows:

1. Everytime the statement references itself or its previous selves, duplicate the statement once but no further for each self reference.
2. If the statement contains modal verbs, negation and any adjectives of some sort, make a new copy of the statement treating those aforme…

Example: Using what the dream said, "This statement is false" will become:

This statement is false
This statement is true

Semiclassical

Something something Yablo paradox

Null

@Secret this statement is true. <- doesn't even make sense

Secret

One of the most hilarious thing is that if you take any pseudoscientific claim as the statement (which probably include a lot of antedotes, figures and stuff and misuse of quotes), at least as seen in the dream it always end up generating an irregular table of size > 10 x 20 something

(Typo: Forgot to type one more thing: For each rule: Whenever a statement is duplicated, the duplicated version no longer has the modifiers that cause it to duplicate in the first place, or it becomes a modifer that does not do not do anything (e.g. true)

so naively speaking it always terminates

MickLH

@Null But this isn't the usual form of not making sense, usually not making sense is something along the lines of no roots, that statement is more like multiple roots in the domain of interest and so you can't know which sense to make of it.

It has two valid interpretations which lead to opposite conclusions :P

Secret

For example: "This is the number 3" gives the table:

This is the number 3
This is the number 2
This is the number 1

Adeek

23:17

@TedShifrin here ?

Null

So: "that is BECAUSE it is written here and there" is a very poor reasoning

Secret

Sorry typo:

This is the number 3 gives:

This is the number 3
This is the number 2
This is the number 2
This is the number 2
This is the number 1
This is the number 1
This is the number 1
This is the number 1
This is the number 1
This is the number 1

MickLH

~~why~~ nvm

mick

Bounty hunters attention !! Placed a bounty here math.stackexchange.com/questions/1553888/…

Hope it Gets an answer

Sophie

that looks hard

Secret

23:30

Q: Is saying 'This statement is true' a logically valid statement?

I understand how 'This statement is false' is not logically valid, but what about 'This statement is true'? I've always heard self-referential statements are not logically sound, but I can't really give a great explanation for why this one would not be. Anyone have a good argument one way or the ...

logic

Null

This is so because it is so?

^how about this one

Secret

I think the outcome is logically equivalent to "This statement is true"
Because assume true, then $T\rightarrow T= T$
Assume it false, $F \rightarrow F = F$

I think it can be more compactly rephrased as "If X then X"

where X is the statement (in this case, X is the statement itself)

Sophie

@mick this looks more like a number theory problem than anything else

Akiva Weinberger

Either this statement is false or you will eat your hat

Sophie

are the residues of the integers equidistributed modulo $\pi$?

Akiva Weinberger

23:43

^ The only way for that not to be paradoxical is for you to eat your hat

@Sophie I believe they're dense

(Doesn't quite answer the question though)

Null

This statement is either true or false.

Akiva Weinberger

That's true; no paradox. (If it's neither then it's false, so it's not neither.)

Null

This statement is either true or false or a unicorn.

Mike Miller

@Sophie yes, this is equivalent to asking whether $k/\pi$ is equidistrubted mod $\Bbb Z$, and that's true of any irrational in place of $1/\pi$

DHMO

x \implies x is a true statement

@Null ^

Null

23:47

^well, one of the first things we proved in our algebra course

Akiva Weinberger

@MikeMiller Why? I feel like it should be easy but I'm missing it

Null

$\neg a\land a\implies$ hilarious

Akiva Weinberger

You want me to pick up waffle cones? Oh, right, for the wine. One sec, let me just derive your son's credit card number and I'll be on my way.

DHMO

"Does Riemann conjecture imply Fermat's Last Theorem?"

mick

Mike Miller

23:49

@AkivaWeinberger I've long forgotten the proof; it's a special case of the ergodic theorem, which I also don't know how to prove

but it's an old theorem so there should be rather elegant and simple proofs

DHMO

yes

@mick it should be yes

Null

i found $\neg a\land a\implies$ indianers at a tipi with guns the funniest, who was it again who brought it up?

DHMO

@Null if 1<0, then P=NP

Akiva Weinberger

Anything implies truth, and falsehood implies anything

DHMO

nice

Akiva Weinberger

23:53

@mick Start by assuming the Riemann Hypothesis. Then mimick the 1995 proof of Fermat's Last Theorem. Conclude.

Null

@AkivaWeinberger mmh

Mike Miller

i wonder if the proof of FLT requires the existence of inaccessible cardinals

DHMO

"If 1=0 then everyone is the pope"

well falsehood implies anything

Akiva Weinberger

@Null If you work in intuitionistic logic, however, you can get cool stuff, like how $p\lor\lnot p$ can't be assumed to be true

It keeps the "falsehood implies anything" thing though

Mike Miller

update: nope, FLT does not require the existence of inaccessible cardinals, and is true just from ZFC

Null

23:58

well, falsehood implies certainly truth

Mike Miller

probably from peano arithmetic but you're going to have a tougher time claiming that they know that as a fact

Null

but some true statements don't imply another true statement I suppose

(read i feel)

Akiva Weinberger

And also, intuitionistic logic includes classical logic (that is, "normal logic"), in a certain sense. Essentially, if you take any classically true statement and put a bunch of $\lnot\lnot$s everywhere, you get something that's true in intuitionistic logic

So in a certain sense intuitionistic logic is richer than classical logic