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Author Topic: Maths thread.  (Read 307160 times)

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Cuzz

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Re: Maths thread.
« Reply #775 on: February 24, 2017, 06:15:54 pm »
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I've just been reading a bunch of Doron Zeilberger's opinions and other writings on ultrafinitism and I legitimately can't tell whether I just find them hilariously snarky or if they're actually starting to sway me.

Wait, how do you know Doron Zeilberger?  He's at Rutgers; I did my PhD there.

Just from legend more or less. There aren't that many ultrafinitists out there.
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Witherweaver

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Re: Maths thread.
« Reply #776 on: February 24, 2017, 06:20:28 pm »
0

I've just been reading a bunch of Doron Zeilberger's opinions and other writings on ultrafinitism and I legitimately can't tell whether I just find them hilariously snarky or if they're actually starting to sway me.

Wait, how do you know Doron Zeilberger?  He's at Rutgers; I did my PhD there.

Just from legend more or less. There aren't that many ultrafinitists out there.

Ah.  He is a quirky guy.  I never did combinatorics so I didn't interact with him much.. I believe I TA'd for him once.
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heron

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Re: Maths thread.
« Reply #777 on: February 24, 2017, 09:53:00 pm »
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I heard about him from one of the mentors at Canada/USA Mathcamp. I think he is fairly widely known (among certain circles).
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Re: Maths thread.
« Reply #778 on: February 24, 2017, 11:00:16 pm »
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I heard about him from one of the mentors at Canada/USA Mathcamp. I think he is fairly widely known (among certain circles).

Oh, you went to Mathcamp? Cool! Mathcamp is actually what got me into Dominion. This was before Isotropic got big, so lots of us did things like open Village/Mining Village.
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SirPeebles

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Re: Maths thread.
« Reply #779 on: February 25, 2017, 11:52:08 am »
+1

I've just been reading a bunch of Doron Zeilberger's opinions and other writings on ultrafinitism and I legitimately can't tell whether I just find them hilariously snarky or if they're actually starting to sway me.

I was going to read his opinions, but man, his list goes on forever.
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sudgy

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Re: Maths thread.
« Reply #780 on: March 04, 2017, 05:24:15 pm »
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I've never taken statistics before, but I feel like the answer to this should be pretty simple.  Say I've done an experiment to determine if a certain value is constant, and I have a bunch of data.  How do I determine the p value for the assumption that it is constant?  Googling just gives me a bunch of other scenarios.
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   Quote from: sudgy on June 31, 2011, 11:47:46 pm

Ratsia

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Re: Maths thread.
« Reply #781 on: March 05, 2017, 02:15:51 pm »
+3

I've never taken statistics before, but I feel like the answer to this should be pretty simple.  Say I've done an experiment to determine if a certain value is constant, and I have a bunch of data.  How do I determine the p value for the assumption that it is constant?  Googling just gives me a bunch of other scenarios.
You're probably not giving enough information here. If everything you have is a set of exchangeable measurements of the value, there's no way of separating measurement noise from changes in the actual value. You need some information about the data points along the aspects you suspect the value might vary (temporal, spatial, or whatever). Whether the answer is simple or not then depends on what kind of dynamics you are looking at as possible alternatives.
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sudgy

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Re: Maths thread.
« Reply #782 on: March 05, 2017, 03:42:53 pm »
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I've never taken statistics before, but I feel like the answer to this should be pretty simple.  Say I've done an experiment to determine if a certain value is constant, and I have a bunch of data.  How do I determine the p value for the assumption that it is constant?  Googling just gives me a bunch of other scenarios.
You're probably not giving enough information here. If everything you have is a set of exchangeable measurements of the value, there's no way of separating measurement noise from changes in the actual value. You need some information about the data points along the aspects you suspect the value might vary (temporal, spatial, or whatever). Whether the answer is simple or not then depends on what kind of dynamics you are looking at as possible alternatives.

It's measuring the equilibrium constant of a chemical reaction with varying initial concentrations of reactants.  I don't know anything about how the concentration would affect it (I know it actually is constant, but this experiment is supposed to be a test of that).  This isn't necessary to do (it's just for a college class), but I thought I could maybe try to be a bit more rigorous in my defense of the hypothesis.
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   Quote from: sudgy on June 31, 2011, 11:47:46 pm

sudgy

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Re: Maths thread.
« Reply #783 on: March 05, 2017, 05:21:02 pm »
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Okay, maybe a p value isn't what I'm looking for, that's just what people usually look for.  My main question is how can I mathematically justify the assumption that my data represents a constant value?
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   Quote from: sudgy on June 31, 2011, 11:47:46 pm

scott_pilgrim

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Re: Maths thread.
« Reply #784 on: March 05, 2017, 05:32:21 pm »
+1

Okay, maybe a p value isn't what I'm looking for, that's just what people usually look for.  My main question is how can I mathematically justify the assumption that my data represents a constant value?

If you're pretty sure that the data should be linear, you can find the least squares regression line (you can probably just google that and find an online tool that will do it for you), and then see how close the slope is to 0 and whether the r^2 value is "good".
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Ratsia

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Re: Maths thread.
« Reply #785 on: March 06, 2017, 01:46:21 am »
+2

It's measuring the equilibrium constant of a chemical reaction with varying initial concentrations of reactants.  I don't know anything about how the concentration would affect it (I know it actually is constant, but this experiment is supposed to be a test of that).  This isn't necessary to do (it's just for a college class), but I thought I could maybe try to be a bit more rigorous in my defense of the hypothesis.
I'd say scott_pilgrim's suggestion is good enough for you. You do have nice pairs of values, the initial concentrations and the actual value, so the question boils down to showing that there is no relationship between the two. By making an assumption that there could be a linear relationship and then showing that the relationship that best fits the data is one with (pretty much) zero slope is what most people would do. Just remember to combine that with a clear plot that shows there are no obvious higher-order relationships -- I guess that kind of a plot is what they are actually looking for as the solution.
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Re: Maths thread.
« Reply #786 on: March 06, 2017, 11:41:14 am »
0

It's measuring the equilibrium constant of a chemical reaction with varying initial concentrations of reactants.  I don't know anything about how the concentration would affect it (I know it actually is constant, but this experiment is supposed to be a test of that).  This isn't necessary to do (it's just for a college class), but I thought I could maybe try to be a bit more rigorous in my defense of the hypothesis.
I'd say scott_pilgrim's suggestion is good enough for you. You do have nice pairs of values, the initial concentrations and the actual value, so the question boils down to showing that there is no relationship between the two. By making an assumption that there could be a linear relationship and then showing that the relationship that best fits the data is one with (pretty much) zero slope is what most people would do. Just remember to combine that with a clear plot that shows there are no obvious higher-order relationships -- I guess that kind of a plot is what they are actually looking for as the solution.

I don't think they're actually looking for anything too fancy (the only math required for this course is algebra).  I was just trying to get extra brownie points or something.
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   Quote from: sudgy on June 31, 2011, 11:47:46 pm

silverspawn

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Re: Maths thread.
« Reply #787 on: March 11, 2017, 06:17:38 am »
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So I finished the set theory lecture a while ago (and read a bit on wikipedia). I got a few questions left, if anyone wants to help. (I got the sense that there are actually a few different models which work slightly differently, so this is about ZFC).

First (this was not really explored in the lecture, I just tried to gather it from thinking & the internet), do I understand it correctly that

ℕ =: ω = {0, 1, 2, ...}

aka the natural numbers and the smallest infinite ordinal. Then the next "bigger" ordinals are

ω+ = ω ∪ {ω} = {0, 1, 2, ..., ω}
ω++ = {0, 1, 2, ..., ω, ω+}
...

and there exists

α := ω ∪ {ω, ω+, ω++, ...} = {0, 1, 2, ..., ω, ω+, ω++, ...}

and then

α+ = α ∪ {α} = {0, 1, 2, ..., ω, ω+, ω++, ... α}
α++ = {0, 1, 2, ..., ω, ω+, ω++, ... α, α+}
...

and there exists

β := {0, 1, 2, ..., ω, ω+, ω++, ... α, α+, α++, ...}

and

γ := {0, 1, 2, ..., ω, ω+, ω++, ... α, α+, α++, ..., β, β+, β++, ...}

and then you can play that game forever and can keep getting ordinals that way, and the ordinal which is the union of all such ordinals is ω1, which is kind of like ℕ x ℕ and therefore still countable in the same way (but the first non-recursive ordinal), so it's still ω1 ~ ℕ and then there exist ω1+, ω1++, ω1+++, ω1 ∪ {ω1, ω1+, ω1++, ...} := Ψ and Ψ+, Ψ++, ω1 ∪ {Ψ, Ψ+, Ψ++}, and this can be done arbitrarily often again, until we get something which is kind of like ℕ x ℕ x ℕ which is still countable, let's call it Φ and then building from Φ you could construct something like ℕ x ℕ x ℕ x ℕ which would still be countable, and if you do that arbitrarily often, then you have the union of all such ordinals, which is also the union of all countable ordinals and is kind of like ℕ x ℕ x ℕ x ℕ ... and called ω0 and the first uncountable ordinal. Is that all correct?

(It's pretty weird. I like it. If it actually works that way.)

Second, if this is all grounded in this set language and can all be done formally, why aren't all mathematical theorems just provable or disprovable through some software which handles First-order logic?

Third, is this in any way applicable beyond providing a generally better understanding of maths? (Not saying this wouldn't be enough reason to study it.)

And fourth, I'm a bit confused about the whole principle of axioms. If the idea is that they give you ways to construct new sets, and your universe always contains exactly all sets which can be constructed this way, doesn't the axiom of foundation (which was the only one introduced which said something doesn't exist) just assert something that must be true anyway (since otherwise it is inconsistent with the other axioms), but for which no-one found a proof (and presumably there is none?) If so, that seems more like a theory that everyone beliefs in rather than an axiom to me, because it's asserting something rather than defining it, and because introducing it doesn't change the universe.

And 4.5, isn't the power set axiom implied by the replacement scheme? It seemed as if otherwise the axioms were meant to have as little redundancy as possible, so that seems odd.

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Re: Maths thread.
« Reply #788 on: March 11, 2017, 09:10:51 am »
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As far as I can tell the replacement scheme does not imply the power set axiom at all. Where did you get that idea?

Without any axioms, we don't know anything about how sets work at all. It would be impossible to do anything with them because  we wouldn't know anything for sure. There are some basic properties about sets which we want to be true because they fit with our intuitive notion for what a set should be and they let us prove lots of other things about sets. These are our axioms. It's not that they haven't been proved; they can't be proved because without them we simply don't know anything. Don't think of them as a belief system or assumptions, rather, they are definitions. The rationale is "let's build math up from sets. Wait, what's a set? Well, let's define it to be a thing which has all these properties..." Many of the axioms relate to which sets exist given that other sets exist, which is certainly important for constructing all the mathematical objects we use every day, but they are really just properties about the sets themselves. For example, the axiom of union says that any set A has the property that we can take a union over it, that is, there exists another set B that contains exactly the elements which are elements of the elements of A.
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Re: Maths thread.
« Reply #789 on: March 11, 2017, 09:41:11 am »
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I don't think you understood the question. I do look at axioms as definitions. The question was, since most axioms expand your universe by giving you tools to construct more sets, why is something that doesn't give you any new sets (the axiom of foundation), but rather asserts that you can't build sets with property X – sets which you cannot construct anyway (otherwise it would contradict your other axioms) – considered an axiom rather than a theorem.

scott_pilgrim

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Re: Maths thread.
« Reply #790 on: March 11, 2017, 10:20:48 am »
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Second, if this is all grounded in this set language and can all be done formally, why aren't all mathematical theorems just provable or disprovable through some software which handles First-order logic?

I thought (someone can correct me if I'm wrong) that while it's possible for a machine to verify an arbitrary proof in first-order logic, it's not possible for a machine to come up with a proof of an arbitrary statement in first-order logic.
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Re: Maths thread.
« Reply #791 on: March 11, 2017, 11:14:02 am »
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I don't think you understood the question. I do look at axioms as definitions. The question was, since most axioms expand your universe by giving you tools to construct more sets, why is something that doesn't give you any new sets (the axiom of foundation), but rather asserts that you can't build sets with property X – sets which you cannot construct anyway (otherwise it would contradict your other axioms) – considered an axiom rather than a theorem.

What contradiction are you talking about? If ZFC is consistent, so is ZFC without foundation.

Regarding automated proofing: ZFC is inconsistent or incomplete (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems), that is there are either theorems that can't be decided, or there's a contradiction in ZFC. All decidable theorems can be proven true or false through some software that handles first-order logic though (but not in finite time), which seems to be kinda tautological to me.

Regarding wether this is is applicable to anything, I don't think so.

There are models of ZF(C) without the power set axiom, which are actually some of the more populary axiom systems known as ZF(C)^{-}.
« Last Edit: March 11, 2017, 11:17:07 am by Watno »
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Watno

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Re: Maths thread.
« Reply #792 on: March 11, 2017, 11:15:41 am »
0

Second, if this is all grounded in this set language and can all be done formally, why aren't all mathematical theorems just provable or disprovable through some software which handles First-order logic?

I thought (someone can correct me if I'm wrong) that while it's possible for a machine to verify an arbitrary proof in first-order logic, it's not possible for a machine to come up with a proof of an arbitrary statement in first-order logic.

Unless I'm mistaken, this is false. If a given statement is decidable, there's a finite proof that it is true or false. Checking all possible proofs will get you to the right one at some point (provided you check them in ascending length).
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silverspawn

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Re: Maths thread.
« Reply #793 on: March 11, 2017, 12:12:33 pm »
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I don't think you understood the question. I do look at axioms as definitions. The question was, since most axioms expand your universe by giving you tools to construct more sets, why is something that doesn't give you any new sets (the axiom of foundation), but rather asserts that you can't build sets with property X – sets which you cannot construct anyway (otherwise it would contradict your other axioms) – considered an axiom rather than a theorem.

What contradiction are you talking about? If ZFC is consistent, so is ZFC without foundation.

Here is what I mean:

assume every axiom except foundation. Now one of those two things must be true

1| based on these axioms, there is a set which violates the axiom of foundation
2| based on these axioms, there is no set which violates the axiom of foundation

If 1| holds, then the axiom of foundation contradicts the other axioms. So I assume 1| doesn't hold. So 2| holds. So there is no set which violates the axiom of foundation – even if you don't assume it. Which begs the question of, why introduce it as an axiom, if it is true anyway? If it is just something that is assumed to be true but cannot be proven, then it doesn't seem to be a definition (hence no axiom) but rather a theorem.

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Re: Maths thread.
« Reply #794 on: March 11, 2017, 02:30:47 pm »
+1

assume every axiom except foundation. Now one of those two things must be true

1| based on these axioms, there is a set which violates the axiom of foundation
2| based on these axioms, there is no set which violates the axiom of foundation
This is false. Just because you can't prove a set exists, that doesn't mean it doesn't.
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Re: Maths thread.
« Reply #795 on: March 11, 2017, 03:06:41 pm »
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I'm not saying you can prove it. I'm just saying either it exists or it doesn't exist.

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Re: Maths thread.
« Reply #796 on: March 11, 2017, 03:56:41 pm »
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In each individual model, yes it does. However, there are models where it does, and models where it doesn't.

Consider for example the theory consisting only of the axiom "Empty sets exists". The model consisting only of a single set is a model of that, but so is any model of ZFC.
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Re: Maths thread.
« Reply #797 on: March 11, 2017, 04:53:42 pm »
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Okay, but there is no model in which such a set exists, is there? It doesn't exist even with all axioms present, so how could it exist with fewer axioms.

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Re: Maths thread.
« Reply #798 on: March 11, 2017, 05:16:32 pm »
+1

Seems to me that you are assuming that sets can only exist if they can be explicitly constructed from the axioms. That's not true at all.
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Re: Maths thread.
« Reply #799 on: March 12, 2017, 01:52:25 am »
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Oh yes, I was assuming that. Isn't that the point of the axiom system, to regulate exactly which sets exist?
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