1724: "Proofs"

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1724: "Proofs"

Postby thunk » Wed Aug 24, 2016 4:05 am UTC

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Alt-text: Next, let's assume the decision of whether to take the Axiom of Choice is made by a deterministic process ...

Now, let's assume that there will be a bunch of commenters who will pull this discussion in a completely unrelated direction...
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Re: 1724: "Proofs"

Postby commodorejohn » Wed Aug 24, 2016 4:35 am UTC

I gotta be honest, this is pretty much how I've always viewed the entire concept of mathematical proofs.
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Re: 1724: "Proofs"

Postby rhomboidal » Wed Aug 24, 2016 4:44 am UTC

Quod Erat Demonicum

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Re: 1724: "Proofs"

Postby serutan » Wed Aug 24, 2016 5:38 am UTC

Ah, reminds me of the college math/physics days when the 2 phrases guaranteed to strike the fear of God in one were "Left as an exercise for the student" and "It is intuitively obvious".
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Re: 1724: "Proofs"

Postby Tub » Wed Aug 24, 2016 7:03 am UTC

serutan wrote:"It is intuitively obvious".

Also known as "you will prove this on your exam". I always wondered what would happen if your exam answer included the words "intuitively" and/or "obvious".

I don't get the complaint though. If we restricted math to statements which our brains are equipped to intuitively understand, we wouldn't get many useful results.

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Re: 1724: "Proofs"

Postby FOARP » Wed Aug 24, 2016 7:20 am UTC

Here's where I have to be honest and say that, despite making it all the way to the end of my degree in physics and astrophysics, I never actually understood anything by looking at the mathematical proofs. I always understood it terms of the underlying logic and viewed the mathematical proof as just being the way of working out the relation between the factors in numerical terms once you understood, logically, what the relationship was.

Of course, this is exactly the approach that got me 12% in a quantum physics exam.

EDIT: the awesome thing was meeting physics profs who basically view the maths used in physics in the same way I did. These guys pretty much seem to rely on algorithms coded by other people though.

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Re: 1724: "Proofs"

Postby orthogon » Wed Aug 24, 2016 10:00 am UTC

serutan wrote:Ah, reminds me of the college math/physics days when the 2 phrases guaranteed to strike the fear of God in one were "Left as an exercise for the student" and "It is intuitively obvious".

Surely most things that are intuitively obvious in Mathematics are wrong. For example, it's intuitively obvious that there are more rationals than integers.

FOARP wrote:Here's where I have to be honest and say that, despite making it all the way to the end of my degree in physics and astrophysics, I never actually understood anything by looking at the mathematical proofs. I always understood it terms of the underlying logic and viewed the mathematical proof as just being the way of working out the relation between the factors in numerical terms once you understood, logically, what the relationship was.

Of course, this is exactly the approach that got me 12% in a quantum physics exam.

EDIT: the awesome thing was meeting physics profs who basically view the maths used in physics in the same way I did. These guys pretty much seem to rely on algorithms coded by other people though.


I feel the same way. If I can see intuitively that something is the case, I might, on a good day, be able to hack the mathematical proof into submission. But that there are people who can play with the maths and come up with an answer that they didn't expect: that is truly remarkable to me.
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Re: 1724: "Proofs"

Postby Neil_Boekend » Wed Aug 24, 2016 10:09 am UTC

FOARP wrote:Here's where I have to be honest and say that, despite making it all the way to the end of my degree in physics and astrophysics, I never actually understood anything by looking at the mathematical proofs. I always understood it terms of the underlying logic and viewed the mathematical proof as just being the way of working out the relation between the factors in numerical terms once you understood, logically, what the relationship was.

Of course, this is exactly the approach that got me 12% in a quantum physics exam.

EDIT: the awesome thing was meeting physics profs who basically view the maths used in physics in the same way I did. These guys pretty much seem to rely on algorithms coded by other people though.

I'm a Bachelor of Engineering and that's basically what I do. I don't do math, I use it.

Sure, I'll add a few numbers without calling in a mathematician. But if you ask me to determine the stability factor of a proposed closed loop amplified measure and control system when you apply a sinus shaped input signal with known frequency and amplitude I'll have to look up the formulas and apply them. I can't even check and verify the results without building said system in real life.
I think the quickest route was a Laplace transform and that would mean it was on page 42 in Appendix A of our textbook. I have predicted systems with Laplace Transforms and then proceeded to build and test them. And they worked as predicted, so I know the Laplace transform technique works. And my maths teacher was very good. He made math fun. And he did explain Laplace transforms, but it didn't stick.

And it works. I don't need to do math in my work. I'm not a mathematician and I'll never be one. But I gladly use the stuff the mathematicians figure out and I'll always respect their work.
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Re: 1724: "Proofs"

Postby cellocgw » Wed Aug 24, 2016 11:13 am UTC

Neil_Boekend wrote:
FOARP wrote:Here's where I have to be honest and say that, despite making it all the way to the end of my degree in physics and astrophysics, I never actually understood anything by looking at the mathematical proofs. I always understood it terms of the underlying logic and viewed the mathematical proof as just being the way of working out the relation between the factors in numerical terms once you understood, logically, what the relationship was.

Of course, this is exactly the approach that got me 12% in a quantum physics exam.

EDIT: the awesome thing was meeting physics profs who basically view the maths used in physics in the same way I did. These guys pretty much seem to rely on algorithms coded by other people though.

I'm a Bachelor of Engineering and that's basically what I do. I don't do math, I use it.
{snip}

And it works. I don't need to do math in my work. I'm not a mathematician and I'll never be one. But I gladly use the stuff the mathematicians figure out and I'll always respect their work.


And us physics students used to laugh at the poor engineering students who couldn't do proofs or derive relations. :P One of our standard insults was along the lines of "hey, whaddya mean that problem was hard? Just look at the same problem in the book with different values for the constants."

Anyway, on a relatively (sorry, Einstein) serious note - the point of a math proof is to show that the conjecture is true, and thus it's safe to use it to calculate other stuff. As opposed to, say, taking the intuitively obvious guess that the harmonic series converges, and getting into all sorts of trouble using said series to get some real-world result wrong.

As the 3-parter goes:

A mathematician assumes a series diverges until proven otherwise.
A physicist assumes a series converges until proven otherwise.
An engineer assumes a series converges even when proven otherwise.
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Re: 1724: "Proofs"

Postby Neil_Boekend » Wed Aug 24, 2016 11:17 am UTC

cellocgw wrote:And us physics students used to laugh at the poor engineering students who couldn't do proofs or derive relations. :P One of our standard insults was along the lines of "hey, whaddya mean that problem was hard? Just look at the same problem in the book with different values for the constants."

I'm quite able to figure out how to rewrite the problem to bring it to a shape where the textbook tools apply so I can use the tools. As should any engineer. But I am unable to create the tools.
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Re: 1724: "Proofs"

Postby doogly » Wed Aug 24, 2016 11:58 am UTC

Regarding something as "just a tool" is a poisonous line of thought.

Except for people who study on Friday nights and tell you to keep quiet, those folks were rightly just tools.
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Re: 1724: "Proofs"

Postby Reecer6 » Wed Aug 24, 2016 12:14 pm UTC

Now, I'm not saying it's not good, but this comic is basically just the recent Business Idea with math instead of business. I cannot wait for 1731: Arcology Blueprints and 1738: Actually a Political Cartoon.

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Re: 1724: "Proofs"

Postby cellocgw » Wed Aug 24, 2016 12:27 pm UTC

Neil_Boekend wrote:{out of context} But I am unable to create the tools.


That's what a RepRap i3 is for. :mrgreen:
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Re: 1724: "Proofs"

Postby richP » Wed Aug 24, 2016 1:00 pm UTC

Ah yes, higher level math classes. "Prove this theorem". What? nah, that's ok professor, I'll take your word for it. After all, *you're* the guy with the PhD. Plus, you seem trustworthy.

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Re: 1724: "Proofs"

Postby Tribunzio » Wed Aug 24, 2016 1:12 pm UTC

Lewis Carrol (fallacies in early attempts at formal logic theory?)
"He thought he saw an Argument
That proved he was the Pope:
He looked again, and found it was
A Bar of Mottled Soap.
'A fact so dread,' he faintly said,
'Extinguishes all hope!'"

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Re: 1724: "Proofs"

Postby Showsni » Wed Aug 24, 2016 1:21 pm UTC

Tribunzio wrote:Lewis Carrol (fallacies in early attempts at formal logic theory?)
"He thought he saw an Argument
That proved he was the Pope:
He looked again, and found it was
A Bar of Mottled Soap.
'A fact so dread,' he faintly said,
'Extinguishes all hope!'"


Should have proved it with a double rule of three.

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Re: 1724: "Proofs"

Postby pscottdv » Wed Aug 24, 2016 1:22 pm UTC

My differential equations class felt this way to me. "Assume a solution of the form..."

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Re: 1724: "Proofs"

Postby 5th Earth » Wed Aug 24, 2016 1:29 pm UTC

Reminds me of the quantum bogosort.
It seemed like a good idea at the time.

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Re: 1724: "Proofs"

Postby Muswell » Wed Aug 24, 2016 4:42 pm UTC

doogly wrote:Regarding something as "just a tool" is a poisonous line of thought.

Except for people who study on Friday nights and tell you to keep quiet, those folks were rightly just tools.


That's a bit harsh, at my university at least some of us had important exams on Saturdays.

People who study on Saturday nights and tell you to keep quite I will grant are tools.

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Re: 1724: "Proofs"

Postby doogly » Wed Aug 24, 2016 5:28 pm UTC

Exams on Saturdays? In what world is this happening?
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Re: 1724: "Proofs"

Postby rmsgrey » Wed Aug 24, 2016 7:34 pm UTC

orthogon wrote:
serutan wrote:Ah, reminds me of the college math/physics days when the 2 phrases guaranteed to strike the fear of God in one were "Left as an exercise for the student" and "It is intuitively obvious".

Surely most things that are intuitively obvious in Mathematics are wrong. For example, it's intuitively obvious that there are more rationals than integers.


It's also intuitively obvious that 4 is bigger than 2.

The thing is, we tend to notice our intuitions being wrong a lot more than we do when they're right, so there's going to be a strong sampling bias here...

In general, I'd say that so long as you restrict yourself to mathing things you're familiar with, your intuitions are going to be fairly sound. When you extend beyond the things with which you're familiar, your intuition becomes rather suspect.

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Re: 1724: "Proofs"

Postby macraw83 » Wed Aug 24, 2016 7:49 pm UTC

doogly wrote:Exams on Saturdays? In what world is this happening?

In my university, the last day of classes was a Wednesday, and then final exam week was generally Friday through Wednesday, including Saturdays and excluding Sundays. They also had a rule where if you had 3 exams starting within a 24-hour period, you could move any one of them to another time, so it was actually nice to be able to spread them out into the weekend some. Also, I never had a Saturday exam start before 10AM or after 2PM, but I may have just been lucky.
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Re: 1724: "Proofs"

Postby Muswell » Wed Aug 24, 2016 8:24 pm UTC

macraw83 wrote:
doogly wrote:Exams on Saturdays? In what world is this happening?

In my university, the last day of classes was a Wednesday, and then final exam week was generally Friday through Wednesday, including Saturdays and excluding Sundays. They also had a rule where if you had 3 exams starting within a 24-hour period, you could move any one of them to another time, so it was actually nice to be able to spread them out into the weekend some. Also, I never had a Saturday exam start before 10AM or after 2PM, but I may have just been lucky.


Start of every term we had exams the Friday before Full Term started to make sure we hadn't been slacking off during the vac; if you had three exams one of them was on Saturday morning, 9.30am. I normally only had two, but got stuck with an extra one a couple of times in my first two years. Those exams didn't count for anything, but tutors could get sarcastic if you did badly.

My second year exams consisted of 10 compulsory and 1 optional 3-hour exams, two a day during the week with the optional one on the Saturday (I did not take this exam; I am not insane). These exams just had to be passed to get through to the second phase of study and did not count for anything long-term. My enduring memories of that week are begging the guys on the floor below me not to have a party, having to walk to exams in a blizzard, and my quarter of the city having a day-long power cut that did not help with revision.

My Finals at the end of 4th year consisted of 8 3-hour exams and two 1.5 hour exams. I was doing a fairly weird combination and got stuck with two Saturday morning sessions, the first one being my first exam and the other being I think my sixth exam. My entire final degree class was based on that week and a half of exams.

Finals were way more spread out than second year exams because at Finals you had a pretty free choice on what topics you did so they had to allow for a lot of combinations, whereas second year exam topics were set in stone other than "pick one topic from list X and one topic from list Y" for two of them, plus the weird optional extra one.

Sometimes I forget why I enjoyed university so much.

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Re: 1724: "Proofs"

Postby Eternal Density » Thu Aug 25, 2016 10:02 am UTC

Neil_Boekend wrote:I'm a Bachelor of Engineering and that's basically what I do. I don't do math, I use it.
I'm fairly similar.

Well actualy, as a software engineer I don't use much math these days. I used to when I did hydrological modelling software though, and I had little to no idea why or how most of the models worked. That was the hydrologists' job.
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Re: 1724: "Proofs"

Postby DanD » Thu Aug 25, 2016 8:47 pm UTC

Eternal Density wrote:
Neil_Boekend wrote:I'm a Bachelor of Engineering and that's basically what I do. I don't do math, I use it.
I'm fairly similar.

Well actualy, as a software engineer I don't use much math these days. I used to when I did hydrological modelling software though, and I had little to no idea why or how most of the models worked. That was the hydrologists' job.


BSME, I have to pull out a few equations every once in a while. But mostly no.

That being said, my favorite joke on mathematical proofs:

A professor, while teaching a class, starts writing out a lengthy and complex proof on the board. The whole time he is writing, he is describing the steps very rapidly. As he seems to be drawing to a close, he comes out with the line "it is therefore obvious..." and trails off. He steps back, blinks a few times, and then runs out of the classroom. The class, very confused, waits around for a while, and eventually filters out.

At the next lecture, a couple of days later, the professor enters the classroom, wearing the same clothes, and with every sign that he has been working steadily in the intervening time. He starts the new lecture with the line "I was right, it is obvious", writes the concluding line, and then continues from there.
Last edited by DanD on Fri Aug 26, 2016 5:10 pm UTC, edited 1 time in total.

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Re: 1724: "Proofs"

Postby Pfhorrest » Thu Aug 25, 2016 9:02 pm UTC

three fellers was tasked to build th' biggest pen they could with as lil fence as possible

one feller who dun engineered things fer a living sez its obvs, jes build it a circle an yer done

nother feller who studied physics sez yeah but th' earth it's round, so cuz a that if ya make it big as possible y'll use less fence per the size of it

third feller builds himself a little tiny pen around him an sez "I'm outside the pen"

i reckon that third feller does math or somesuch

get it?
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Re: 1724: "Proofs"

Postby jewish_scientist » Fri Aug 26, 2016 12:32 am UTC

Pfhorrest wrote:three fellers was tasked to build th' biggest pen they could with as lil fence as possible

This gave me flashbacks to me calculus final. The question was the same except one side was a river, so it did not need any fencing. After expending all the testing time and working on it outside the professor's office while he prepared to leave for the day, the answer I got was something like a rectangle measuring 40,000/infinitesimal feet by infinity feet. :|
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Re: 1724: "Proofs"

Postby pelrigg » Fri Aug 26, 2016 1:12 am UTC

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Re: 1724: "Proofs"

Postby Pfhorrest » Fri Aug 26, 2016 2:24 am UTC

jewish_scientist wrote:
Pfhorrest wrote:three fellers was tasked to build th' biggest pen they could with as lil fence as possible

This gave me flashbacks to me calculus final. The question was the same except one side was a river, so it did not need any fencing. After expending all the testing time and working on it outside the professor's office while he prepared to leave for the day, the answer I got was something like a rectangle measuring 40,000/infinitesimal feet by infinity feet. :|

Mind you I never did actually finish calculus so I'm only second-handedly familiar with the concepts thereof, but if it's really the same question (get the highest area-enclosed-per-length-of-fence ratio) but with the river defining one edge, wouldn't the correct answer just be to build a semicircular enclosure? Discounting the line used to bisect the circle (thanks to that being a river), a semicircle has the same perimeter-to-area ratio as a circle (half the perimeter and half the area of the bisected circle), so if a circle is the optimal solution sans river, a semicircle would be optimal with river, no?
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Re: 1724: "Proofs"

Postby Eternal Density » Fri Aug 26, 2016 3:01 am UTC

Pfhorrest wrote:three fellers was tasked to build th' biggest pen they could with as lil fence as possible

one feller who dun engineered things fer a living sez its obvs, jes build it a circle an yer done

nother feller who studied physics sez yeah but th' earth it's round, so cuz a that if ya make it big as possible y'll use less fence per the size of it

third feller builds himself a little tiny pen around him an sez "I'm outside the pen"

i reckon that third feller does math or somesuch

get it?
My thoughts on the matter:
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Re: 1724: "Proofs"

Postby thunk » Fri Aug 26, 2016 4:57 am UTC

Pfhorrest wrote:third feller builds himself a little tiny pen around him an sez "I'm outside the pen"

You mean an ink pin...why not use a sewing pin?
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Re: 1724: "Proofs"

Postby rmsgrey » Fri Aug 26, 2016 11:09 pm UTC

Pfhorrest wrote:
jewish_scientist wrote:
Pfhorrest wrote:three fellers was tasked to build th' biggest pen they could with as lil fence as possible

This gave me flashbacks to me calculus final. The question was the same except one side was a river, so it did not need any fencing. After expending all the testing time and working on it outside the professor's office while he prepared to leave for the day, the answer I got was something like a rectangle measuring 40,000/infinitesimal feet by infinity feet. :|

Mind you I never did actually finish calculus so I'm only second-handedly familiar with the concepts thereof, but if it's really the same question (get the highest area-enclosed-per-length-of-fence ratio) but with the river defining one edge, wouldn't the correct answer just be to build a semicircular enclosure? Discounting the line used to bisect the circle (thanks to that being a river), a semicircle has the same perimeter-to-area ratio as a circle (half the perimeter and half the area of the bisected circle), so if a circle is the optimal solution sans river, a semicircle would be optimal with river, no?


Is the river straight? Does it matter? (answer: yes - if the river meanders in such a way that two loops are separated by the available length at their closest approach, then a locally optimal solution is a straight fence along that line)

But, yeah, one proof that the closed curve containing maximum area (in Euclidean space) is a circle starts by proving that the optimal curve is made up of two curves, each of which encloses the maximum possible area when closed by connecting the endpoints with a straight line. So, with a straight boundary that's as long as you need, the optimum shape is indeed a semi-circle.

Of course, the actual question may have specified additional constraints - for example, that the enclosure must be a rectangle - which would change the answer.

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Re: 1724: "Proofs"

Postby xtifr » Sat Aug 27, 2016 1:03 am UTC

Hmm, for some reason, this conversation is reminding me of a series of jokes my dad and some of his nerdy friends used to make up when I was a wee 'un.

The topic was: how would someone of profession X find a sheep. Possible answers included: bombarding the earth with anti-sheep, and looking for an explosion (experimental physicist), or climbing in a sack and inverting the universe (topologist), or build a fence bisecting the earth, inspect to determine which side has the sheep, build a new fence bisecting that portion, and repeat until the sheep is found (computer scientist). I wish I could remember more of them.
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Re: 1724: "Proofs"

Postby SuicideJunkie » Sat Aug 27, 2016 10:54 am UTC

The topic was: how would ... professor X find a sheep.
Well, I suspect he'd fire up Cerebro to search for the evil cloning facility, and then send in the team to fight the monstrous creations before they can transform all of Europe's population into wool sock-puppets.

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Re: 1724: "Proofs"

Postby Copper Bezel » Sat Aug 27, 2016 11:35 am UTC

rmsgrey wrote:Of course, the actual question may have specified additional constraints - for example, that the enclosure must be a rectangle - which would change the answer.

It changes it in the sense that you now need half of the rectangular shape that has least perimeter per area, rather than the half of the most area-optimized shape outright, but it's not that great a leap from half a circle to half a square and doesn't make it any less funny when someone ends up dividing by infinity instead.
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Re: 1724: "Proofs"

Postby rmsgrey » Sat Aug 27, 2016 1:54 pm UTC

Copper Bezel wrote:
rmsgrey wrote:Of course, the actual question may have specified additional constraints - for example, that the enclosure must be a rectangle - which would change the answer.

It changes it in the sense that you now need half of the rectangular shape that has least perimeter per area, rather than the half of the most area-optimized shape outright, but it's not that great a leap from half a circle to half a square and doesn't make it any less funny when someone ends up dividing by infinity instead.


It does make solving it by calculus a lot easier.

And, yeah, having infinite and infinitesimal side-lengths suggests that they tried constraining to a fixed area rather than a fixed length, and then found the maximum length rather than the minimum length...

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Re: 1724: "Proofs"

Postby Pfhorrest » Sat Aug 27, 2016 4:28 pm UTC

Copper Bezel wrote:
rmsgrey wrote:Of course, the actual question may have specified additional constraints - for example, that the enclosure must be a rectangle - which would change the answer.

It changes it in the sense that you now need half of the rectangular shape that has least perimeter per area, rather than the half of the most area-optimized shape outright, but it's not that great a leap from half a circle to half a square and doesn't make it any less funny when someone ends up dividing by infinity instead.

Actually wouldn't requiring it be rectangular create the exact kind of infinite/infinitesimal problems originally raised? Let's start with a 1:2 rectangle (half a square) with one of its long sides defined by the river. Same fence/area ratio as a square. Now if we make it 1:4 instead we get 200% the area for only 150% the fence. If we double it again we get 200% the area again for only 133% the fence. Double it again and we get 200% the area for 125% the fence. Again to get 200% for 120%. Every doubling gets you the same additional area for less additional fence so why not continue it to infinity?
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Re: 1724: "Proofs"

Postby orthogon » Sat Aug 27, 2016 7:50 pm UTC

To work as an optimisation problem the question has to ask for the maximum area for a given length of fence. Asking for the maximum area per unit length of fence is going to go to infinity whatever shape you make it, because the area scales with the square of the length. This is the sort of thing that explains why you can't have insects the size of elephants and suchlike. But if the fence is of fixed length, you'll end up with a square pen, and as rmsgrey points out, the calculus is dramatically simpler, there being only one independent variable, and well within the reach of A-level maths students or equivalent. Left as an exercise to the reader, all the same ;-)
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Re: 1724: "Proofs"

Postby somitomi » Sat Aug 27, 2016 9:38 pm UTC

Tub wrote:
serutan wrote:"It is intuitively obvious".

Also known as "you will prove this on your exam". I always wondered what would happen if your exam answer included the words "intuitively" and/or "obvious".

I don't get the complaint though. If we restricted math to statements which our brains are equipped to intuitively understand, we wouldn't get many useful results.

I can think of a whole load of these:
  • "As you have already learned in the earlier semesters"
  • "The proof can be found in the lecture notes"
  • "It is trivial"
  • "[finishes writing an equation taking up the entire width of the board] We can clearly see, that..."
Someone suggested a game of bingo, where the card would be filled with all sorts of expressions the lecturer will likely say and anyone acieving a bingo would announce it by yelling "BULLSHIT". Of course the "inventor" takes no responsibility for any consequences of actually playing this game.
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Re: 1724: "Proofs"

Postby Pfhorrest » Sun Aug 28, 2016 12:06 am UTC

orthogon wrote:To work as an optimisation problem the question has to ask for the maximum area for a given length of fence. Asking for the maximum area per unit length of fence is going to go to infinity whatever shape you make it, because the area scales with the square of the length. This is the sort of thing that explains why you can't have insects the size of elephants and suchlike. But if the fence is of fixed length, you'll end up with a square pen, and as rmsgrey points out, the calculus is dramatically simpler, there being only one independent variable, and well within the reach of A-level maths students or equivalent. Left as an exercise to the reader, all the same ;-)

As I said I never actually finished calculus so apologies if this is a stupid question, but I was only talking about scaling the height/width ratio of the rectangle, not actually doubling it's size, so e.g. double the half square you start with and then scale down to a size such that the perimeter is the same as you started, repeat ad infinitum. Since scaling shouldn't affect the area/perimeter ratio (e.g. a circle with perimeter x will always have an area of pi(x/2pi)^2; the ratio of x to that will vary with x but we're talking about keeping x fixed) then it still seems like stretching your rectangle out as skinny as possible buys you the most area provided you don't run out of river to supply you with free perimeter. Why doesn't it?
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