Dogma in Math

For the discussion of math. Duh.

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ahammel
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Re: Dogma in Math

Point the first: code tags.

Point the second: you should be thinking about what your function returns in edge cases like n <= 0 because, sure as shit, in a real program somebody will try feeding it a number less than zero, or a float, or a string, and that person is going to hate you when it silently returns the wrong answer instead of failin noisily. And that person is probably going to be you.
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frog42
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Re: Dogma in Math

ahammel wrote:Point the first: code tags.

Point the second: you should be thinking about what your function returns in edge cases like n <= 0 because, sure as shit, in a real program somebody will try feeding it a number less than zero, or a float, or a string, and that person is going to hate you when it silently returns the wrong answer instead of failin noisily. And that person is probably going to be you.

Points taken, however the course-provided program fails as spectacularly as mine for negative numbers (i.e. returns 1 every time), and while mathematically incorrect, anyone who doesn't have the benefit of this thread's information will input factorial(0) and be left scratching their heads.

It's only an exercise, but if they had an expectation for the output of 0 bricks, they should have provided that information beforehand.

ahammel
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Re: Dogma in Math

frog42 wrote:Points taken, however the course-provided program fails as spectacularly as mine for negative numbers (i.e. returns 1 every time), and while mathematically incorrect, anyone who doesn't have the benefit of this thread's information will input factorial(0) and be left scratching their heads.
If you think fact(0) should be undefined, it's your job as a programmer to make your program express that fact. It's no good complaining that the spec wasn't explicit enough if your program is buggy. The spec is never explicit enough.
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capefeather
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Re: Dogma in Math

I already had a headache from browsing through this and seeing user Eugene_Nier's Christian conservative spam X( I applaud people for taking this guy so seriously and genuinely trying to help, but I honestly feel like you guys just gave him a fish. I would agree that teaching him to fish - getting him to shed his ego and learn how to actually do math and programming - would be a long and arduous process. In which case, here's a YouTube channel.

ahammel
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Re: Dogma in Math

Responding to math and programming questions with nothing but a link to a YouTube channel? Dick move.
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Re: Dogma in Math

Give a man a program and he'll be frustrated for a day. Teach a man to program and he'll be frustrated for life.

frog42
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Re: Dogma in Math

capefeather wrote:I already had a headache from browsing through this and seeing user Eugene_Nier's Christian conservative spam X( I applaud people for taking this guy so seriously and genuinely trying to help, but I honestly feel like you guys just gave him a fish. I would agree that teaching him to fish - getting him to shed his ego and learn how to actually do math and programming - would be a long and arduous process. In which case, here's a YouTube channel.

It was closer to having a fish thrown at me over and over, then eventually someone realizing I ordered steak.

Your alternative is for me to dig through boring, non-interactive youtube channels, trying to fast forward to the part that is pertinent to my question? Perhaps wait days or weeks for a reply to my question in a comment, then try to hold a conversation in a format not designed for such? Running a marathon barefoot through broken glass is fine, but I'd prefer to take a slightly less masochistic approach. Learning through social interaction seems like a pretty solid method to me.

frog42
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Re: Dogma in Math

ahammel wrote:
frog42 wrote:Points taken, however the course-provided program fails as spectacularly as mine for negative numbers (i.e. returns 1 every time), and while mathematically incorrect, anyone who doesn't have the benefit of this thread's information will input factorial(0) and be left scratching their heads.
If you think fact(0) should be undefined, it's your job as a programmer to make your program express that fact. It's no good complaining that the spec wasn't explicit enough if your program is buggy. The spec is never explicit enough.

Dude... I'm on Lesson 3 of Intro to CS. No need to go all drill sergeant on me yet. The exercises right now are to focus on some new, small tidbit and encourage thinking about how problems can be broken out and described. Best practices will have to wait until I've actually learned a few practices.

tooyoo
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Re: Dogma in Math

frog42 wrote:Dude... I'm on Lesson 3 of Intro to CS. No need to go all drill sergeant on me yet. The exercises right now are to focus on some new, small tidbit and encourage thinking about how problems can be broken out and described. Best practices will have to wait until I've actually learned a few practices.

In my experience, the maths examples used as exercises to teach programming can be confusing since they get introduced for the sake of exercise without any context. Maybe it's a bit like teaching someone who's never seen a house how to lay bricks.

Schrollini
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Re: Dogma in Math

frog42 wrote:Dude... I'm on Lesson 3 of Intro to CS. No need to go all drill sergeant on me yet.

Admittedly, I've never served in the army, but aren't drill sergeants let loose on the new recruits right away? They don't wait for them to reach PFC before sending in the drill sergeants. This, I presume, is to ensure that the recruits don't have a chance to pick up any bad habits. That seems completely applicable here.
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ahammel
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Re: Dogma in Math

frog42 wrote:Dude... I'm on Lesson 3 of Intro to CS. No need to go all drill sergeant on me yet.

And now not only have you learned how to write a factorial function, you've also learned about the empty product, why factorial(0) = 1, and that, unless you are very careful, edge cases like that will bite you.

Sounds like a pretty good course.
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gmalivuk
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Re: Dogma in Math

frog42 wrote:Are we allowed to swear in here? Too late, I guess.
Swearing itself is not a problem, but being a rude shitstain is.

frog42 wrote:Dude... I'm on Lesson 3 of Intro to CS. No need to go all drill sergeant on me yet.
When you start a thread as such an antagonistic yet ignorant asshole, you don't get to whine when people are harsh right back at you.
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kubit
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Re: Dogma in Math

frog42 wrote:I'm using parentheses around nothing to represent an empty set, which I believe (and could be mistaken) is unique from the "number" zero.

Actually, in the set theoretic construction of the natural numbers, the empty set is usually considered to be the number zero and the first ordinal number. And this construction is probably the most sensible mathematical construction of the natural numbers there is.

To be honest you could just as well be asking 'why do we define anything to be as it is?' and the only answer is that we try and define things in ways that are most convenient. You don't WANT to leave something undefined like 1/0 for example. Thing is some things are just too difficult to give a sensible definition. 0! does have a perfectly good candidate for definition. For certain things such as 0^0 the definition may vary depending on what is useful...

Yakk
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Re: Dogma in Math

You might want to step back a bit.

Math isn't some "here is the truth from on high". Math is a bunch of symbol manipulation following arbitrary rules that we have found *useful*.

Now, sometimes the kind of *useful* it is is really removed from counting sheep, especially as you reach the more abstract maths. Other times it is not.

The factorial is useful for a bunch of purposes. What 0! is ends up being a function of what would it be useful for it to be, historically, and (eventually) in the future. There are lots of patterns that make 0!=1 both predictable and useful, so that is how we define it.

0! = 1 because of the gamma function being useful, because it makes some combinatorics functions generate useful stuff, etc.

0^0 = 1 often because X^Y where X and Y are sets being the set of functions from the set Y to the set X, and when X and Y are natural numbers being the cardinality of said functions. At other times, 0^0 = 0, because we have a function f_y(x) = y^x and we want it to be continuous in x for all y>=0 and do not mind it being discontinuous in y when x=0.

It isn't dogma that makes 0!=1. It might be taught to you as dogma because experience has shown educators that if you start with foundational concepts in mathematics, you lose 90% of the class before you get to teaching the times tables. By teaching mathematics dogmatically, the "more useful applied" stuff sticks (times tables, that A choose B = A!/(B!(A-B)!), standard deviations, etc) even if the student never learns the foundations. If you teach foundations, they might understand the concrete stuff better when they finally get to it, but they also might be completely lose long before you reach any concrete stuff.

On the other hand, this could simply be because elementary schools are chock full of math fearing educators, and when asked to teach abstract foundational mathematics end up freezing up. When asked to teach rote mathematics, they do not. What more, parents are capable of helping kids with the rote mathematics they grew up on, while having a parent help with a 3rd grade text book that teaches naive set theory is going to be unusual.

I don't know the exact historical pedagogy that resulted in the "teach kids abstract mathematics" failing, but I have read some of the the "new math" textbooks that tried it, and it appears we have since abandoned it. Anyone know more?

Oh, and such a rant is incomplete without a link to Lockhart's famous essay:
http://www.maa.org/sites/default/files/ ... Lament.pdf
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

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Re: Dogma in Math

To add to phlip's excellent post:
It may be helpful to also look at the reverse process - see what happens when we remove sequence elements from the running total and running product. When we add {2, 3, 4}, we get a total of 9. If we subtract those numbers in any order from 9 we eventually get back to zero. Similarly, if we divide 24 by {2, 3, 4} in any order we eventually get back to one.

On factorial in Python:
There's a factorial function in the math module; it throws appropriate exceptions when fed bad arguments.
But here's a simple factorial function that does rudimentary bad arg handling:

Code: Select all

`def fac(n):    if n<0:         raise ValueError, 'factorial() not defined for negative values'    return reduce(lambda u,v:u*v, xrange(1, n+1), 1)`

kubit wrote:Actually, in the set theoretic construction of the natural numbers, the empty set is usually considered to be the number zero and the first ordinal number. And this construction is probably the most sensible mathematical construction of the natural numbers there is.

Quite.

Thus:
0 {}
1 {{}}
2 {{}, {{}}}
3 {{}, {{}}, {{}, {{}}}}
4 {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}
5 {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}}
etc.
The above was created using:

Code: Select all

`#!/usr/bin/env python''' Ordinal sets '''import sysdef numsets(n):    z = []    for i in range(n):        f = "{%s}" % repr(z)[1:-1].replace("'","")        print i, f        z += [f]def main():    n = int(sys.argv[1]) if len(sys.argv) > 1 else 5    numsets(n + 1)if __name__ == '__main__':    main()`

FWIW, I originally wrote that program a few years ago when I was first learning Python, but I cleaned it up slightly this evening.

And finally, here's an Euler diagram for the ordinal 10 that I created several years before that, writing in the PostScript language.

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Re: Dogma in Math

PM 2Ring wrote:And finally, here's an Euler diagram for the ordinal 10 that I created several years before that, writing in the PostScript language.

You could turn that into wallpaper and pretend it's the 1970s again.
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Re: Dogma in Math

besides the fact that many before me have pointed out, that mathematics is an invention and not a discovery, and as such is defined at our will in a way that it is consistent, also with statements such as n/0=⊥ (because division is the inverse of multiplication: find a number which you can multiply to 0 and have n!=0), I also don't see why 0! should be different than 1.
What kind of questions are you trying to answer asking yourself what's 0! ?
If you are asking yourself "How many possible configurations of the state there can be with no elements?" you'll agree that it is the same number as if there is one element, which is 1.

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Re: Dogma in Math

I can't really speak much more to the topic, since I think it's been covered nicely here (and I agree pretty much with what phlip, Yakk, PM 2Ring and others have said about it). I will say to the OP that it's great that you've come to understand this concept even though it's not strictly part of your course, and I hope that it is the beginning of many such understandings about how, out of the infinite variety of ways mathematicians could choose to do things, they went the way they did. I also hope that the next time you do it in a way that is less adversarial - perhaps more "I don't understand this, can someone help?" and less "This is stupid, what idiot would do things this way?".
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Re: Dogma in Math

This is making me think about some sort of dogme 95 for mathematics.

1. Shooting must be done on location. Props and sets must not be brought in (if a particular prop is necessary for the story, a location must be chosen where this prop is to be found).
2. The sound must never be produced apart from the images or vice versa. (Music must not be used unless it occurs where the scene is being shot.)
3. The camera must be hand-held. Any movement or immobility attainable in the hand is permitted.
4. The film must be in colour. Special lighting is not acceptable. (If there is too little light for exposure the scene must be cut or a single lamp be attached to the camera).
5. Optical work and filters are forbidden.
6. The film must not contain superficial action. (Murders, weapons, etc. must not occur.)
7. Temporal and geographical alienation are forbidden. (That is to say that the film takes place here and now).
8. Genre movies are not acceptable.
9. The film format must be Academy 35 mm.
10. The director must not be credited.

These are the rules for dogme 95. What would the rules be for maths dogma 2013? 10 would be "no author", or possibly "the author is always Bourbaki", 8 would maybe be some sort of restriction on the set of axioms. 9 is maybe "produced as a Latex Report" with no author inserted typesetting.

Edit: Thinking about it now, dogma 95 was about cutting through the hype and fame and irreality that surrounds the film industry. There isn't enough hype in maths to make Wiles, or Perelman or anyone else who's done something amazing lately. Perhaps math dogma should be about creating hype.
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frog42
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Re: Dogma in Math

ConMan wrote:I can't really speak much more to the topic, since I think it's been covered nicely here (and I agree pretty much with what phlip, Yakk, PM 2Ring and others have said about it). I will say to the OP that it's great that you've come to understand this concept even though it's not strictly part of your course, and I hope that it is the beginning of many such understandings about how, out of the infinite variety of ways mathematicians could choose to do things, they went the way they did. I also hope that the next time you do it in a way that is less adversarial - perhaps more "I don't understand this, can someone help?" and less "This is stupid, what idiot would do things this way?".

I don't think I could have asked for things to turn out better, and I actually attribute that somewhat to my initial tone. It forced those responding to assume an oppositional stance dedicated to proving me incorrect, which requires straight-forward, explicit reasoning. I find that heated argument (if you can remain rational) is one of the faster ways to really understand someone else. It almost always ends with: "Why are we still yelling when we're pretty much saying the same damn thing at this point?!"

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Re: Dogma in Math

frog42 wrote:...

I am male, I am 'him'.

Yakk
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Re: Dogma in Math

And it gets you added to ignore lists. /bye!
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

ahammel
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Re: Dogma in Math

frog42 wrote:I don't think I could have asked for things to turn out better, and I actually attribute that somewhat to my initial tone.
Yeah, well, thanks for making it clear that it's your policy to be a dick when asking for help.
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Re: Dogma in Math

frog42 wrote:I don't think I could have asked for things to turn out better, and I actually attribute that somewhat to my initial tone. It forced those responding to assume an oppositional stance dedicated to proving me incorrect, which requires straight-forward, explicit reasoning. I find that heated argument (if you can remain rational) is one of the faster ways to really understand someone else. It almost always ends with: "Why are we still yelling when we're pretty much saying the same damn thing at this point?!"
Yeah, no, you'll get an oppositional stance dedicated to proving you incorrect whenever people think you're incorrect. This is the internet after all. So when you decide on top of that to intentionally enter a discussion as a hostile asshole, it's not the fastest way to understand the opposing view, it's just trolling.
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Re: Dogma in Math

frog42, I guess if you don't mind that the majority of the thread thought you were an idiot and treated you as such, then by all means, go for it. You probably would have gotten your desired answer much sooner if you hadn't triggered the hostilities of people who otherwise could have provided it (which is a good fraction of this forum, I'd wager).
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tooyoo
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Re: Dogma in Math

Uh. It doesn't really matter, but since people seem to be lining up at the end of this thread to tell frog how apparently inappropriate his way of questioning was: I don't agree with what people are writing here. His initial posts contains the obvious clumsy, ill-conceived, and ill-advised attempt at adding some humour to his question by referring to uses of LEGO. But, at least where I come from, that is still quite removed from behaviour that qualifies you as a "hostile asshole", "a dick", "a rude shitstain", or an "antagonistic yet ignorant asshole" (or behaviour that gets you banned or ignored).

It was a bad attempt at being flippant, but still a very obvious attempt at that. So maybe we shouldn't all line up to punish the kid, if all he's guilty of is an inability to be funny.

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Re: Dogma in Math

frog42 wrote:
ConMan wrote:I can't really speak much more to the topic, since I think it's been covered nicely here (and I agree pretty much with what phlip, Yakk, PM 2Ring and others have said about it). I will say to the OP that it's great that you've come to understand this concept even though it's not strictly part of your course, and I hope that it is the beginning of many such understandings about how, out of the infinite variety of ways mathematicians could choose to do things, they went the way they did. I also hope that the next time you do it in a way that is less adversarial - perhaps more "I don't understand this, can someone help?" and less "This is stupid, what idiot would do things this way?".

I don't think I could have asked for things to turn out better, and I actually attribute that somewhat to my initial tone. It forced those responding to assume an oppositional stance dedicated to proving me incorrect, which requires straight-forward, explicit reasoning. I find that heated argument (if you can remain rational) is one of the faster ways to really understand someone else. It almost always ends with: "Why are we still yelling when we're pretty much saying the same damn thing at this point?!"

I disagree with your idea that promoting a hostile tone increases the quality of the discussion.
For example, promoting a hostile tone in this recent discussion made for terrible conversation.

Personally, I would have preferred everyone to calmly explain the error of your ways to you, instead of reacting in a hostile way to your suggesting that reacting in a hostile way is good. You're not setting a good example for our Frog buddy here, above posters!

z4lis
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Re: Dogma in Math

It was a legitimate question wrapped up with some obnoxious statements that I imagine the OP will be embarrassed to have written in 5-6 years. I have my fair share of such posts, no doubt...

At any rate, if a whole bunch of people tell you that your behavior isn't acceptable in their community, you should probably own up to your lack of judgement and not do it again, rather than claim you were trying to provoke "better discussion". It, in fact, brings about much worse discussion. For instance, I will never debate with someone who I don't think will actually listen to what I'm saying. Your first posts made me think you were one of those people.
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Re: Dogma in Math

tooyoo wrote:Uh. It doesn't really matter, but since people seem to be lining up at the end of this thread to tell frog how apparently inappropriate his way of questioning was: I don't agree with what people are writing here. His initial posts contains the obvious clumsy, ill-conceived, and ill-advised attempt at adding some humour to his question by referring to uses of LEGO. But, at least where I come from, that is still quite removed from behaviour that qualifies you as a "hostile asshole", "a dick", "a rude shitstain", or an "antagonistic yet ignorant asshole" (or behaviour that gets you banned or ignored).

It was a bad attempt at being flippant, but still a very obvious attempt at that. So maybe we shouldn't all line up to punish the kid, if all he's guilty of is an inability to be funny.

I totally would've ignored the Lego brick stupidity if he hasn't just told us that he intentionally antagonizes the people he's asking for help in order to provoke "better discussion". Where I'm from, we call that 'trolling'.
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frog42
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Re: Dogma in Math

I don't recall suggesting that I intentionally antagonized any of you. I took a radical stance and threw some bravado behind it. Many of you seem rather fast to yell "troll" at every damn situation. If I'd been putting you all on, it'd be one thing, but I wasn't brought around until phlip's post. Calling someone with conviction a troll actually makes you kind of an asshole.

Did I not ultimately accept a well-reasoned argument? If I antagonized anyone, it was individuals who were giving me circular logic (i.e. "because it makes this other formula work"). I might have gotten the "answer" faster if I'd approached as a humble petitioner, but I can't imagine it would have been ingrained as deeply. Look at it this way: By approaching the question from an obstinate (but not irrational), opposing position, more energy is required to reach the conclusion. More energy requires more thought. More thought causes more connections to be formed. Learning requires struggle and understanding, otherwise it's in one ear and out the other.

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Re: Dogma in Math

You could also read the forums in a smaller font or sent through a machine translation and back, if your goal was to throw obstacles up in your own path because you can't be bothered to think things through unless it's difficult for you.

And your question was answered immediately with no circularity and with direct reference to your own plastic-up-the-butt thought experiment: There is exactly one way to stick zero LEGOs up your butt (namely, don't stick any in there), so it makes sense that 0!=1,by your own criterion about what the function is supposed to mean.
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ahammel
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Re: Dogma in Math

I don't have a problem with your "conviction", if that's what you're pleased to call it. I would've given you the benefit of the doubt for your hostile and abrasive tone as well: plenty of people are irritating in ignorance. But when it was pointed out to you that you were antagonizing the people who were taking the time to help you, your only reply was to say that you were pleased with the results, and that you think you get better discussion that way.

Bully for you, but I certainly won't make the mistake of reading your posts anymore, if that's your attitude.
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frog42
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Re: Dogma in Math

gmalivuk wrote:You could also read the forums in a smaller font or sent through a machine translation and back, if your goal was to throw obstacles up in your own path because you can't be bothered to think things through unless it's difficult for you.

And your question was answered immediately with no circularity and with direct reference to your own plastic-up-the-butt thought experiment: There is exactly one way to stick zero LEGOs up your butt (namely, don't stick any in there), so it makes sense that 0!=1,by your own criterion about what the function is supposed to mean.

If that rings of logic to you, you're a better man than I. Accepting that is some Buddhist Koan-level shit. What is the sound of 0! hands clapping?

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Re: Dogma in Math

frog42 wrote:
gmalivuk wrote:You could also read the forums in a smaller font or sent through a machine translation and back, if your goal was to throw obstacles up in your own path because you can't be bothered to think things through unless it's difficult for you.

And your question was answered immediately with no circularity and with direct reference to your own plastic-up-the-butt thought experiment: There is exactly one way to stick zero LEGOs up your butt (namely, don't stick any in there), so it makes sense that 0!=1,by your own criterion about what the function is supposed to mean.

If that rings of logic to you, you're a better man than I. Accepting that is some Buddhist Koan-level shit. What is the sound of 0! hands clapping?

So what you're saying is, you STILL don't understand what has been explained so far?

"Better learning though hard work" means you, the learner, should work hard. Not that you should be obstinate and abrasive to those attempting to explain something to you.
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frog42
Posts: 52
Joined: Wed Aug 26, 2009 8:35 am UTC

Re: Dogma in Math

ahammel wrote:I don't have a problem with your "conviction", if that's what you're pleased to call it. I would've given you the benefit of the doubt for your hostile and abrasive tone as well: plenty of people are irritating in ignorance. But when it was pointed out to you that you were antagonizing the people who were taking the time to help you, your only reply was to say that you were pleased with the results, and that you think you get better discussion that way.

Bully for you, but I certainly won't make the mistake of reading your posts anymore, if that's your attitude.

My reply was that I thought my absorption was better than if I had attempted a more humble supplication. Not that it was better discussion. Did you somehow infer that I'm always going to use this same tactic? Did I not make repeated posts showing my gratitude? Hell, you didn't even show up until the whole thing was over, so I'm not sure how you feel slighted?

somehow
Posts: 99
Joined: Wed Aug 14, 2013 8:31 pm UTC

Re: Dogma in Math

I suspect the LEGOs-up-the-butt thing isn't really going to get anyone anywhere, but since you mention it: if you're opposed to the idea that there's one way to stick no LEGOs up your butt, how many ways do you think there are? Are you saying there are no ways? To me it seems that "There are no ways to do X" is equivalent to "X is impossible", which certainly isn't the case here...
Xenomortis wrote:O(n2) takes on new meaning when trying to find pairs of socks in the morning.

frog42
Posts: 52
Joined: Wed Aug 26, 2009 8:35 am UTC

Re: Dogma in Math

doogly wrote:So what you're saying is, you STILL don't understand what has been explained so far?

"Better learning though hard work" means you, the learner, should work hard. Not that you should be obstinate and abrasive to those attempting to explain something to you.

I'm saying I accept the mathematical reasoning, but that it doesn't sound logical. It doesn't track with linguistics.

frog42
Posts: 52
Joined: Wed Aug 26, 2009 8:35 am UTC

Re: Dogma in Math

somehow wrote:I suspect the LEGOs-up-the-butt thing isn't really going to get anyone anywhere, but since you mention it: if you're opposed to the idea that there's one way to stick no LEGOs up your butt, how many ways do you think there are? Are you saying there are no ways? To me it seems that "There are no ways to do X" is equivalent to "X is impossible", which certainly isn't the case here...

Fine, let's make a nice G-Rated version. How many ways can you stack zero books? 0!=1. So there is one way to stack zero books. However, this doesn't involve any stacking. So you haven't actually stacked anything. So the question defies logic.

frog42
Posts: 52
Joined: Wed Aug 26, 2009 8:35 am UTC

Re: Dogma in Math

Alternatively:

-Call the police! I've killed a man one way!

*Police arrive*

-So how did you kill him?

-I didn't.

This sounds stupid, but when you're high on Math, it's normal.

somehow
Posts: 99
Joined: Wed Aug 14, 2013 8:31 pm UTC

Re: Dogma in Math

I wasn't objecting to the non-G-rated nature of the LEGOs-up-the-butt example.

Sure, the question "how many ways are there to stack zero books?" doesn't have a lot of bearing on the real world, and as such is not really something one would sensibly ask in actual conversation. (I assume that's more or less what you mean by "It doesn't track with linguistics"?) This is not an argument against asking it in a mathematical context. We ask plenty of mathematical questions which seem silly or nonsensical if you try to understand them as non-mathematical language, and that doesn't make them any less legitimate as mathematical questions.

I guess really what I'm saying is that thinking about this in terms of LEGOs or books or whatever real-world objects you prefer only gets you so far. It's a great analogy, most of the time. What you're pointing out is essentially that question "How many ways are there to stack n books?" (arguably) becomes physically meaningless for n = 0. Sure. If you don't like the idea of there being one way to stack zero things, fine. But that's not a sign that 0! = 1 doesn't make sense; after all, the factorial function is an abstract mathematical object that exists quite independently from this whole stacking question. Rather, it's a sign that thinking of n! in terms of possible ways to stack n things doesn't make sense when n = 0, and you need to come at 0! from some other angle. (Like, for example, the ones provided by others earlier in the thread.)
Xenomortis wrote:O(n2) takes on new meaning when trying to find pairs of socks in the morning.