1856: "Existence Proof"

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1856: "Existence Proof"

Postby Raidri » Wed Jun 28, 2017 3:18 pm UTC

Image

Alt-text: "Real analysis is way realer than I expected."

Looks more like complex analysis to me. And I'm not sure which part is real and which is imaginary ...

PS: That last quote definitely needs an exclamation mark instead of a period: "I'm finally in the right one!"

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Re: 1856: "Existence Proof"

Postby jozwa » Wed Jun 28, 2017 3:25 pm UTC

Raidri wrote:PS: That last quote definitely needs an exclamation mark instead of a period: "I'm finally in the right one!"

I couldn't agree more.

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Re: 1856: "Existence Proof"

Postby orthogon » Wed Jun 28, 2017 3:30 pm UTC

Is she suffering from existential angst?
xtifr wrote:... and orthogon merely sounds undecided.

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Re: 1856: "Existence Proof"

Postby sonar1313 » Wed Jun 28, 2017 3:37 pm UTC

Of course, in that dream everyone always has, I didn't read the syllabus and now I have to go home and get my sword, and by the time I get back, the whole class will be gone.

P.S. I like the period better than the exclamation point.

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Re: 1856: "Existence Proof"

Postby Reka » Wed Jun 28, 2017 3:53 pm UTC

I want to be in this math class.

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Re: 1856: "Existence Proof"

Postby cellocgw » Wed Jun 28, 2017 3:57 pm UTC

Is that the same sword used when she was on the thesis defense committee?
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Re: 1856: "Existence Proof"

Postby Flumble » Wed Jun 28, 2017 4:02 pm UTC

Raidri wrote:Looks more like complex analysis to me. And I'm not sure which part is real and which is imaginary ...

Find its conjugate, bring them together and the real part comes out twice. It's a happy little family.

Or find Euler's constant (probably buried with him) and raise it to an x. Once it's grown up, its size will be proportional to x's real part and it will be oriented at x's imaginary part. That only really works if x isn't too imaginary, though.


Does an x with x^2=-1 really exist?

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Re: 1856: "Existence Proof"

Postby svenman » Wed Jun 28, 2017 4:56 pm UTC

orthogon wrote:Is she suffering from existential angst?

Looks to me like she is successfully dealing with it.

And thus is born the category of destructive proofs.

ETA:
Flumble wrote:Does an x with x^2=-1 really exist?

That, of course, depends on how you define "really exist"...
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Re: 1856: "Existence Proof"

Postby cellocgw » Wed Jun 28, 2017 5:03 pm UTC

Flumble wrote:Does an x with x^2=-1 really exist?


Does "2" really exist?
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Re: 1856: "Existence Proof"

Postby Reka » Wed Jun 28, 2017 5:05 pm UTC

cellocgw wrote:Does "2" really exist?

Of course it does, and 2+2=5... for sufficiently large values of 2.

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Re: 1856: "Existence Proof"

Postby qvxb » Wed Jun 28, 2017 5:30 pm UTC

Kim Kardashian possesses sufficiently large values of 2, as does Jenny McCarthy, et all.

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Re: 1856: "Existence Proof"

Postby crystalmeph » Wed Jun 28, 2017 5:33 pm UTC

That number's name is bleem. You will not find it so easy to defeat, I am afraid. It has learned to hide itself from those who would do it harm, and fight back against those who persist in hunting it.

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Re: 1856: "Existence Proof"

Postby jc » Wed Jun 28, 2017 5:46 pm UTC

orthogon wrote:Is she suffering from existential angst?

You'd expect that to be an occupational hazard of mathematicians. After all, their work deals mostly with things that have no actual physical existence, in this universe or any other.

OTOH, there are a number of common jokes among mathematicians about a colleague who thinks his entire life work has been destroyed by a scientists or (even worse) an engineer that just announced a practical application for his theorems.

I've seen claims that George Boole had a bit of a reaction like this when his "purely theoretical" system like the real numbers except for having two operators that each distribute over the other was found to have actual applications. Dunno if it's true, though, or someone was just having a bit of fun with the idea.

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Re: 1856: "Existence Proof"

Postby JohnTheWysard » Wed Jun 28, 2017 5:49 pm UTC

Shouldn't that be

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Re: 1856: "Existence Proof"

Postby Steve the Pocket » Wed Jun 28, 2017 9:18 pm UTC

I like that "I think I'm in the wrong class" wasn't the last line in the comic. Once upon a time it might have been, but Randall knows his audience very well at this point, and knows that there would inevitably be someone like us in that class.
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Re: 1856: "Existence Proof"

Postby keldor » Thu Jun 29, 2017 12:14 am UTC

JohnTheWysard wrote:Shouldn't that be

"Grab your surds, students; we ride!"


Yes!

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Re: 1856: "Existence Proof"

Postby keldor » Thu Jun 29, 2017 12:19 am UTC

She should be careful. There exists f such that there are infinitely many values of x that meet that condition. We wouldn’t want to ride out against those, now would we?

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Re: 1856: "Existence Proof"

Postby zjxs » Thu Jun 29, 2017 12:52 am UTC

Calculators, mount up!
Last edited by zjxs on Thu Jun 29, 2017 11:10 pm UTC, edited 1 time in total.

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Re: 1856: "Existence Proof"

Postby chridd » Thu Jun 29, 2017 1:03 am UTC

...why does my number line have a hole between 1 and 3? Why is there no integer between them, but two integers' worth of non-integer numbers?
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Re: 1856: "Existence Proof"

Postby Pfhorrest » Thu Jun 29, 2017 3:07 am UTC

chridd wrote:...why does my number line have a hole between 1 and 3? Why is there no integer between them, but two integers' worth of non-integer numbers?

What do you mean? I count |R| non-integers between 1 and 3, just like between any two consecutive integers.
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Re: 1856: "Existence Proof"

Postby chridd » Thu Jun 29, 2017 3:21 am UTC

Really? I count 2|ℝ| integers in that interval. And why is there both 1½ and 1 3/2 (which is different from 3½)? Why is the number of fractions with any particular denominator on the interval [1,3) about twice what it is on intervals [0,1), [3,4), [4,5), etc.?
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Re: 1856: "Existence Proof"

Postby Pfhorrest » Thu Jun 29, 2017 3:31 am UTC

2|R| = |R| (that's the joke)

The rest of those are good questions, though.
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Re: 1856: "Existence Proof"

Postby da Doctah » Thu Jun 29, 2017 7:23 am UTC

svenman wrote:
Flumble wrote:Does an x with x^2=-1 really exist?

That, of course, depends on how you define "really exist"...


I once had to take an incomplete in a philosophy class. I made a mistake in a homework assignment and accidentally proved my professor didn't exist.

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Re: 1856: "Existence Proof"

Postby svenman » Thu Jun 29, 2017 9:07 am UTC

da Doctah wrote:I once had to take an incomplete in a philosophy class. I made a mistake in a homework assignment and accidentally proved my professor didn't exist.

That sounds like another thing that could have been fixed by judicious application of a sword. :twisted:
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Re: 1856: "Existence Proof"

Postby somitomi » Thu Jun 29, 2017 9:29 am UTC

da Doctah wrote:I once had to take an incomplete in a philosophy class. I made a mistake in a homework assignment and accidentally proved my professor didn't exist.

As long as the proof is consistent, everything is all right.
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Re: 1856: "Existence Proof"

Postby Xenomortis » Thu Jun 29, 2017 12:02 pm UTC

Flumble wrote:Does an x with x^2=-1 really exist?

Yeah, but it's not unique.
Image

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Re: 1856: "Existence Proof"

Postby orthogon » Thu Jun 29, 2017 1:53 pm UTC

Xenomortis wrote:
Flumble wrote:Does an x with x^2=-1 really exist?

Yeah, but it's not unique.

Breaking News: mathematicians have just discovered that the number they thought was i may in fact have been i's evil twin, −i, all along. The two are not identical, although at the time of writing, nobody can tell which is which.

A spokesperson for the IEEE was relaxed about the possible case of mistaken identity. "We've been using j anyway," they said, "so we can simply define j=−i if the flat-footed mathematicians decide they'd had the wrong guy all this time."
xtifr wrote:... and orthogon merely sounds undecided.

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Re: 1856: "Existence Proof"

Postby DavidSh » Thu Jun 29, 2017 2:34 pm UTC

Xenomortis wrote:
Flumble wrote:Does an x with x^2=-1 really exist?

Yeah, but it's not unique.

What do we have? i, -i, j, -j, k, and -k ?

Or x could be 1 in the field of integers modulo 2.
orthogon wrote:Breaking News: mathematicians have just discovered that the number they thought was i may in fact have been i's evil twin, −i, all along. The two are not identical, although at the time of writing, nobody can tell which is which.

A spokesperson for the IEEE was relaxed about the possible case of mistaken identity. "We've been using j anyway," they said, "so we can simply define j=−i if the flat-footed mathematicians decide they'd had the wrong guy all this time."


But that would set ij = 1, while we know ij = k.

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Re: 1856: "Existence Proof"

Postby heiligerdankgesang » Thu Jun 29, 2017 2:42 pm UTC

She should have been wielding the Axe of Choice.

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Re: 1856: "Existence Proof"

Postby Flumble » Thu Jun 29, 2017 3:18 pm UTC

DavidSh wrote:
Xenomortis wrote:
Flumble wrote:Does an x with x^2=-1 really exist?

Yeah, but it's not unique.

What do we have? i, -i, j, -j, k, and -k ?

We can do better than that.

orthogon wrote:mathematicians have just discovered that the number they thought was i may in fact have been i's evil twin, −i, all along.

Oh no! Not em-dash i!

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Re: 1856: "Existence Proof"

Postby orthogon » Thu Jun 29, 2017 3:47 pm UTC

Flumble wrote:
orthogon wrote:mathematicians have just discovered that the number they thought was i may in fact have been i's evil twin, −i, all along.

Oh no! Not em-dash i!

Anticipating such a comment, I deliberately copied and pasted it from the Wikipedia page for "Plus and minus signs". :(
xtifr wrote:... and orthogon merely sounds undecided.

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Re: 1856: "Existence Proof"

Postby Flumble » Thu Jun 29, 2017 5:12 pm UTC

orthogon wrote:
Flumble wrote:
orthogon wrote:mathematicians have just discovered that the number they thought was i may in fact have been i's evil twin, −i, all along.

Oh no! Not em-dash i!

Anticipating such a comment, I deliberately copied and pasted it from the Wikipedia page for "Plus and minus signs". :(

I shouldn't have assumed a long horizontal line is an em- or en-dash in this context.
Today I learned there are quite a few of them ﹣-᠆‐–‑−‒﹘—―─

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Re: 1856: "Existence Proof"

Postby Darkhand » Fri Jun 30, 2017 5:45 am UTC

So math me up... She's saying that there should be a number x that simultaneously equals 0 and 1 when the gravitational constant and this mystery function are involved?

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Re: 1856: "Existence Proof"

Postby orthogon » Fri Jun 30, 2017 6:32 am UTC

Darkhand wrote:So math me up... She's saying that there should be a number x that simultaneously equals 0 and 1 when the gravitational constant and this mystery function are involved?

Really they're two separate equations: f(x)=1 and G(f(0))=1. However, only the former says anything about x. The latter isn't itself a condition that can be satisfied or not depending on the value of x. It's presumably simply being asserted as true, perhaps as a result of the earlier parts of the derivation, or by definition, in the manner that one might say "f(x)=√81=9".

Using capital letters to denote functions isn't unusual, but there's normally something different about them: e.g. in Fourier analysis, capital letters are used for frequency-domain functions and lower case for time domain.
xtifr wrote:... and orthogon merely sounds undecided.

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Re: 1856: "Existence Proof"

Postby Soupspoon » Fri Jun 30, 2017 12:05 pm UTC

The way I have read it, "eff of x and Gee of <whatever the eff of zero is> both equals one". In and of itself, this is an open-ended equation that is easily satisfied by any f() or G() functions you might wish1. We just don't know how we got to this point in the lesson, with prior equations... And it looks like some of the students absolutely weren't too sure where it was going, either...

(Different fields of mathematics/sciences have different reasons to use capitalised function-letters, another unknown. But <constant G> times a function something would normally be G.f(x), give or take order preferences and maybe other field-specific uses of the dot-multiplier that might encourage or discourage another mark or perhaps allow none.)

More esoteric mathematical fields may understand it totally differently, though. People do all kinds of strange things to represent the odder algebras. ;)


1 Try G(x)=x or G(x)=x² (amongst many others, including plain …=1), and any of various f()s and x values that happen to make f(x)=f(0)=1, regardless of what else it does (such as f(x)=1-x itself for the ultra-special case x=0).

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Re: 1856: "Existence Proof"

Postby orthogon » Fri Jun 30, 2017 12:49 pm UTC

Soupspoon wrote:The way I have read it, "eff of x and Gee of <whatever the eff of zero is> both equals one". In and of itself, this is an open-ended equation that is easily satisfied by any f() or G() functions you might wish1. We just don't know how we got to this point in the lesson, with prior equations... And it looks like some of the students absolutely weren't too sure where it was going, either...

Me too; when I said there were two equations, I meant that both were simultaneously true. However, the claim is that there exists some x that satisfies them both, and since G(f(0)) is not a function of x, G(f(0))=1 must simply be gnomically true.

Mathematical notation is highly ambiguous, and heavily dependent on some very arbitrary conventions. A lower-case letter followed by something in parentheses would normally be a function, e.g. f(x+1), but if you see a lower-case x or y, it probably is multiplying the contents of the parentheses, e.g. x(y+1). So it depends on which letter is used, with f,g and h the go-to letters for functions and x,y and z for variables. When these conventions are violated it can really mess with the reader's ability to follow the mathematics, in the way that prose written in an unfamiliar dialect might be hard to follow. My personal dislikes are using superscripts for indices as opposed to exponentiation (Einstein apparently came up with this for his tensor notation, but he, unlike me, was clever enough to handle it) and reversing the role of main symbol and subscript, e.g. AT to mean "absolute time" instead of TA. I can see the temptation for speakers of adj-noun languages, but it's just wrong.

Mathematical notation is quite an interesting phenomenon in linguistic terms, since it has elements of natural language about it.
xtifr wrote:... and orthogon merely sounds undecided.

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Re: 1856: "Existence Proof"

Postby chridd » Fri Jun 30, 2017 8:54 pm UTC

Soupspoon wrote:"eff of x and Gee of <whatever the eff of zero is> both equals one"
I initially read that as "whatever the f*** zero is".
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Re: 1856: "Existence Proof"

Postby Soupspoon » Fri Jun 30, 2017 9:18 pm UTC

chridd wrote:
Soupspoon wrote:"eff of x and Gee of <whatever the eff of zero is> both equals one"
I initially read that as "whatever the f*** zero is".

And so you might! Have you never tried dividing by it? There's something odd about nothing, for sure! (Except that it's definitely even... Gah!)

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Re: 1856: "Existence Proof"

Postby rmsgrey » Fri Jun 30, 2017 10:38 pm UTC

Soupspoon wrote:
chridd wrote:
Soupspoon wrote:"eff of x and Gee of <whatever the eff of zero is> both equals one"
I initially read that as "whatever the f*** zero is".

And so you might! Have you never tried dividing by it? There's something odd about nothing, for sure! (Except that it's definitely even... Gah!)


And then there's one, which is definitely odd - how can it be a number with no multiplicity to it? And it's neither prime nor composite, more evidence of it not really being a number...


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