Dogma in Math

For the discussion of math. Duh.

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Forest Goose
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Re: Dogma in Math

Postby Forest Goose » Tue Nov 12, 2013 6:52 am UTC

The number of permutations of a set X is |X|!

The Cartesian product of the empty set with itself is the empty set.

The only subset of the empty set is the empty set, this subset vacuously satisfies the conditions that it is a function.

This subset also satisfies, vacuously, that it is a permutation.

Thus, there is at least 1 permutation empty -> empty and there is no more than 1; thus, there is exactly 1.

By the first line, 0! = 1.
-----

Please point out the error

*Also, as for the acting like a jerk thing, I don't see how that helps you at all. You sort of sound like you're a step away from denying the empty set exists. Perhaps you should study more before being combative and making sweeping declarations of "dogma" in a subject you don't seem to understand.
Last edited by Forest Goose on Tue Nov 12, 2013 6:54 am UTC, edited 1 time in total.
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Re: Dogma in Math

Postby jestingrabbit » Tue Nov 12, 2013 6:53 am UTC

frog42 wrote: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.


"the number of ways in which you could order the people I've killed is 1".
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Re: Dogma in Math

Postby frog42 » Tue Nov 12, 2013 7:03 am UTC

somehow wrote: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.)


Yes, I get that. And many people in this thread have liked to say "it's not about counting sheep". But is that true? Ultimately, isn't all math about counting sheep? Even if you get into the completely abstract, theoretical realms, won't the foundation still be sheep? We may cut the sheep out of the equation because they can mess things up, but can you ever actually get away from them? *Asking sincerely, not rhetorically.*

The "not counting sheep" trope comes off a bit pretentious to me. Like abstract art.

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Re: Dogma in Math

Postby z4lis » Tue Nov 12, 2013 7:07 am UTC

I don't think you're understanding the purpose of a definition. It doesn't make sense to "stack 5 books in no ways" any more than it makes sense to "multiply 2 by itself negative one-half times". And yet, you probably don't have any problems with the symbols [imath]2^{-\frac{1}{2}}[/imath] defined to mean [imath]\frac{1}{\sqrt{2}}[/imath]. As you already probably know, it makes sense to interpret weird things like "multiply a number by itself some horrible number of ways" to be such that the following formulas hold: [imath]a^{b+c} = a^ba^c, a^{cb} = (a^b)^c[/imath]. These formulas hold when the definitions make perfectly good sense, and it turns out they can be meaningfully extended to domains under which the original interpretation is nonsensical. But we always have to be careful with such things. For instance, when I write [imath](-1)^\frac{1}{2}[/imath], do I mean [imath]i[/imath] or [imath]-i[/imath]?

Another example: 1 + 2 + 3 + 4 + ... = ?. Of course, nobody actually thinks that if you add up all the integers, you get a finite number. The point is that there is a formula that makes perfect sense on some domain that can be meaningfully extended off of the domain. If you try to "reinterpret" the extension, you can get weird things. So it's not always valid, but still useful.

Another example: In calculus, you will almost never run into trouble just taking [imath]\frac{1}{\infty} = 0[/imath]. But this formula, formally, makes no sense. [imath]\infty[/imath] isn't a number. You can't divide by it. But that's still a useful formula, since if I take the limit of two functions and formally derive the LHS, then the limit is the RHS. But we don't make a definition like 0/0 = 1. Why not this? It almost never works! It would be a useless definition that makes the symbols more difficult and confusing to work with, not less.

This is what is going on with factorial. The number of size k subsets of a set of size n is given by [imath]\frac{n!}{(n-k)!k!}[/imath]. But there is a subset of size n, the whole set, and a subset of size 0, the empty set. So the formula should spit out 1 in each of these cases, and if we simply define 0! = 1, it works out. And no other issues arise, in almost any imaginable circumstance. It's consistent and useful notation. That's why we make the definition. If you still have a problem with the fact we take 0! = 1 after reading this, then you would probably have better luck taking your problem to a philosophy forum, because your criticisms probably won't be mathematically founded.

EDIT: I also highly suggest you formulate an actual problem you're facing. So far, your only argument seems to be "well, I still don't like it".
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Re: Dogma in Math

Postby frog42 » Tue Nov 12, 2013 7:16 am UTC

At this point, it seems all we have left to discuss is philosophy, and since that is not the reason for this forum, I think I'll be moving on.

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Re: Dogma in Math

Postby Forest Goose » Tue Nov 12, 2013 7:27 am UTC

frog42 wrote:At this point, it seems all we have left to discuss is philosophy, and since that is not the reason for this forum, I think I'll be moving on.


You're just outright rejecting anything contrary to what you say, that's not "all that's left to discuss is philosophy" that's "I refuse to accept anything except what I declare". What I'm confused about is your rejections, you're being given mathematical arguments, you're responding with "I don't find that intuitive"- what I don't understand is why what you find intuitive matters? A lot of mathematics is brain breakingly counter intuitive, that's not a very good reason to reject it, it's a good reason to assume you don't have good intuition.

It's one thing to ask for help figuring out how to get an intuition for something, it's another thing to dismiss something because other's can't make it intuitive for you. The first is great, the latter is arrogant. --And pretentious, as was your comment about sheep counting and pretentiousness.
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Re: Dogma in Math

Postby PM 2Ring » Tue Nov 12, 2013 7:36 am UTC

frog42 wrote:I will tentatively not label you a brick fucker. Are we allowed to swear in here? Too late, I guess.

Hi frog42, welcome to the xkcd forums. Yes, you are allowed to swear here. That's not the problem.

At xkcd, we don't care what people do with Lego bricks in the privacy of their own homes. However, insulting someone for unusual practices that you personally find distasteful is frowned upon here, as is attempting to insult people by implying that they engage in such practices. (OTOH, there is a restricted part of the forum where you can engage with consenting people in all sorts of no-holds-barred forum games...)

HTH

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Re: Dogma in Math

Postby dudiobugtron » Tue Nov 12, 2013 8:08 am UTC

frog42 wrote: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.

Why would you say you killed a man, if you killed 0 men?

frog42 wrote:What is the sound of 0! hands clapping?

I laughed at this. :)
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Re: Dogma in Math

Postby mat.tia » Tue Nov 12, 2013 10:23 am UTC

frog42 wrote:
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.

dude just stop sticking with this question of stacking things.
you have an environment. and n objests that can be placed in different order on a line. How many ways can the environment look if you have n elements? n!. How many ways can it look if it has 0 elements? 1. => 0!=1.
You pretend maths works according to how you phrase things and how you picture them. If things don't match either 1) math is wrong 2) your question is wrong or at least is not appropriate to be related to that problem

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Re: Dogma in Math

Postby curtis95112 » Tue Nov 12, 2013 2:15 pm UTC

frog42 wrote:I've just started studying programming, and one assignment was to create a program that would return the factorial of any positive integer. So n!=x. Here's where I ran into what I consider dogma. The test of the program included 0. And supposedly, 0!=1... This makes no sense. 1!=1... Are you trying to say, through the law of equivalency, that 0!=1! ?

I can find no logical reason for this, other than laziness. n/0 = error, so why do we give special privilege to 0! ? Why not simply make 0!=( ) ?

Is it not time for us to evolve away from viewing zero as a number? It seems like we have to bend over backwards to make zero as a number fit with our worldview, and what are we really getting in return?

My ultimate argument: For the set of all the LEGO bricks you've inserted into your anus, how many different ways can we stack those bricks?

Now accepting any logical argument from the gallery of LEGOphiles.


Putting the lego issue to one side, what the hell is wrong with "Why in the world would 0!=1?". Do you sincerely believe that this would have triggered an inferior discussion?
Instead, you decided to assert that everyone was lazy and bending over backwards. You also tried to make your question sound sophisticated by invoking the "law of equivalency", which isn't a standard term for anything as far as I'm aware. And despite being a self-professed rank amateur, you grandly proclaimed it was time to "evolve away from viewing zero as a number". Keep in mind that that phrase told practically all of us that you didn't know what numbers, zero, or evolution is.

frog42 wrote:I've just started studying programming, and one assignment was to create a program that would return the factorial of any positive integer. So n!=x.
The test of the program included 0. And supposedly, 0!=1... This makes no sense. 1!=1... Are you trying to say, that 0!=1! ?

I can find no logical reason for this. n/0 = error, so why do we give special privilege to 0! ? Why not simply make 0!=( ) ?


This question would have been received much better. And the quality of the discussion wouldn't have suffered at all.

Now, to content.

If you're still uncomfortable with permutations of the empty set, then forget about it. The factorial function is a function from the integers to the integers. It's something we've defined for our convenience. We could define 0! = -1, and it still wouldn't be wrong. But as it turns out, defining 0! = 1 turns out to be the most convenient. For instance nCn = n!/(n!(n-n)!), wouldn't work otherwise and we'd have to write in an exception. Note that nothing of substance would change either way. The content of maths is independent of the notation we use for it.

In short, you're arguing way out of your depth about something of merely superficial importance and being a huge dick about it.

I recommend reading a textbook on set theory. That's probably the best way to understand how empty sets work in modern mathematics. Comfort with 0! = 1 will come naturally.

EDIT: Apparently the law of equivalency refers to the principle of "an eye for an eye" in law.
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Re: Dogma in Math

Postby lorb » Tue Nov 12, 2013 3:27 pm UTC

curtis95112 wrote:
frog42 wrote: And supposedly, 0!=1... This makes no sense. 1!=1... Are you trying to say, through the law of equivalency, that 0!=1! ?

You also tried to make your question sound sophisticated by invoking the "law of equivalency", which isn't a standard term for anything as far as I'm aware.


By context I assume the "law of equivalency" frog42 refers to is the transitivity of the equality relation. Or in simpler terms: a=b and c=b implies a=c, thus 0!=1 and 1!=1 implies 0!=1!
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Re: Dogma in Math

Postby mike-l » Tue Nov 12, 2013 4:04 pm UTC

lorb wrote:
curtis95112 wrote:
frog42 wrote: And supposedly, 0!=1... This makes no sense. 1!=1... Are you trying to say, through the law of equivalency, that 0!=1! ?

You also tried to make your question sound sophisticated by invoking the "law of equivalency", which isn't a standard term for anything as far as I'm aware.


By context I assume the "law of equivalency" frog42 refers to is the transitivity of the equality relation. Or in simpler terms: a=b and c=b implies a=c, thus 0!=1 and 1!=1 implies 0!=1!



Huh, I assumed he was referring to Kabbalah

Anyway, from
-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.

It's clear you still don't understand. In context, not stacking anything is not a way of stacking 5 books. It's that the empty stack is a way (the only way) of stacking the empty set.
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Re: Dogma in Math

Postby curtis95112 » Tue Nov 12, 2013 5:22 pm UTC

lorb wrote:
curtis95112 wrote:
frog42 wrote: And supposedly, 0!=1... This makes no sense. 1!=1... Are you trying to say, through the law of equivalency, that 0!=1! ?

You also tried to make your question sound sophisticated by invoking the "law of equivalency", which isn't a standard term for anything as far as I'm aware.


By context I assume the "law of equivalency" frog42 refers to is the transitivity of the equality relation. Or in simpler terms: a=b and c=b implies a=c, thus 0!=1 and 1!=1 implies 0!=1!


I agree. My point was that frog was feigning sophistication by making terms up (Just in case it wasn't clear).
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Tyndmyr wrote:
Роберт wrote:Sure, but at least they hit the intended target that time.

Well, if you shoot enough people, you're bound to get the right one eventually.

Thats the best description of the USA ever.

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Re: Dogma in Math

Postby Forest Goose » Tue Nov 12, 2013 8:10 pm UTC

frog42 wrote: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.


Math, the new bath salts: you'll get so high you can't even properly incriminate yourself- Just Say No to the empty set!
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Re: Dogma in Math

Postby Gwydion » Tue Nov 12, 2013 11:09 pm UTC

frog42 wrote: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.
Maybe the better analogy would be the following: Imagine a set of timelines in which n people are murdered. In each timeline, the murders were committed in a different order (not simultaneous). How many possible timelines can exist?

If there are 3 corpses, there are 3! = 6 possible timelines. If there are 2, there are 2! = 2 timelines. If there is one corpse, there can only be one timeline. However, if there are 0 dead bodies, there is still one timeline - the one in which everybody is alive.

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Re: Dogma in Math

Postby dudiobugtron » Wed Nov 13, 2013 1:39 am UTC

jestingrabbit wrote: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.


I think we missed an opportunity here. Allow me to attempt to rectify that.

6) The paper must not contain superficial results. (Lemmas, corollaries, etc. must be rigorously proved.)
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Re: Dogma in Math

Postby mike-l » Wed Nov 13, 2013 4:19 pm UTC

Gwydion wrote:
frog42 wrote: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.
Maybe the better analogy would be the following: Imagine a set of timelines in which n people are murdered. In each timeline, the murders were committed in a different order (not simultaneous). How many possible timelines can exist?

If there are 3 corpses, there are 3! = 6 possible timelines. If there are 2, there are 2! = 2 timelines. If there is one corpse, there can only be one timeline. However, if there are 0 dead bodies, there is still one timeline - the one in which everybody is alive.


Or, simply, "I've killed no men in one way" which makes perfect sense, there's only one way to kill noone, and that's to kill noone. Killing noone in 0 ways means you haven't killed noone, ie you've killed at least one person.
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Re: Dogma in Math

Postby doogly » Wed Nov 13, 2013 4:26 pm UTC

"I just can't imagine any ways to kill no people," he said, biting his nails. "You really mean, there is way to kill... nobody? I mean, I think I know what you're saying, but it just doesn't feel right... there is a way?"

He looked over his shoulder nervously, then returned with a steely gaze. "No, what you're saying is just dogma. Everyone tells you such a life is possible, but you know it to be nonsense, deep down, don't you? Killing no people... it doesn't really make sense. Sure, it may be convenient for the kind of life you live to pretend there is a way..."
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Re: Dogma in Math

Postby FancyHat » Wed Nov 13, 2013 4:29 pm UTC

mike-l wrote:Or, simply, "I've killed no men in one way" which makes perfect sense, there's only one way to kill noone, and that's to kill noone. Killing noone in 0 ways means you haven't killed noone, ie you've killed at least one person.

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