Best Religious/Nonreligious Debate Ever

[_marg]

Stak, Marg etc.
Don't miss the post (above) that I made late last night concerning mathematical induction etc.

I just realized you resurrected this thread. I'm only on page 2 atm..and I have no time this afternoon..it will take me some time to catch up...but I'm glad to see you are involved..because just Stak and I discussing goes nowhere.

[_MrStakhanovite]

They are not always numbered the same way so you must specify. Are you talking about the axiom of induction?

As I understand it, the axiom of induction would be:

If X(0) is true and X(n) -> X(n+1) for all n then X is true for all n

If X is a property of a given set (for example), I don't understand why this has to be considered deduction, because there would be more data in the conclusion, than in the conjunction of stated premises.

by the way, I automatically concede the issue to you, but I’m self taught here and since I’m screwing this up, I hope I can turn this into a quick lesson.

[_marg]

Tarski, I've read the whole thread. To simply so as to not quote and go through another laborious process assuming you've read the thread could you please tell me if was right or wrong in my arguments..that Dawkins was not making a deductive argument and that Stak was mixed up in his understanding of validity. Please give your general overall assessment of that discussion between Stak and myself. One of the reasons is Stak claimed I enter threads unprepared, don't know what I'm talking about etc. and used this thread as an example. So I'd like to know which one of us in the thread seemed to not know what they were talking about. thanks

[Guest]

by the way Tarski there is also this thread in which we continued discussing the same issue ..it appears your recent posts are saying the same thing I was back then to Stak.

viewtopic.php?p=360962#p360962

Show me what "forms" there are in inductive reasoning Stak.

You are right Marg, Form never matters in inductive reasoning...you can move any of those numbers around with any signs and the result is always the same!

Why do you assume that is inductive reasoning? The speculation initially may be consider inductive..but that proof is deductive reasoning.


Here you go Stak:

http://en.wikipedia.org/wiki/Mathematical_induction

Mathematical induction should NOT be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics (see Problem of induction for more information). In fact, mathematical induction is a form of rigorous deductive reasoning.

And by the way..you haven't taken a first year logic course... amirite?... as you would say...hence you're none response to my question on that.

And this was his response to me:

Why do you assume that is inductive reasoning?

Stak: Gee, I don't know Marg, maybe because it's trying to make a statement about all natural numbers.

The speculation initially may be consider inductive..but that proof is deductive reasoning.

Stak: Marg you are a dumb human being. No really, you are a dense, slow witted, person who kinda sucks at life. I'm amused the first source you finally bring up in Wikipedia, but let me put this into a better context for you:

Principle of Mathematical Induction (Mathematics)


Show true for n = 1
Assume true for n = k
Show true for n = k + 1
Conclusion: Statement is true for all n >= 1

The key word in step 2 is assume. You are not trying to prove it's true for n = k, you're going to accept on faith that it is, and show it's true for the next number, n = k + 1. If it later turns out that you get a contradiction, then the assumption was wrong.

Stak: What people do with Mathematical induction (And it is reasoning, all math is reasoning) is form a conjecture and assume it's true till proven otherwise. They turn that conjecture into an axiom (2nd order) and deduce from there. So, when you say:

The speculation initially may be consider inductive..but that proof is deductive reasoning.

Stak: That is literally one of the dumbest things you could possibly say. All it is doing is saying " okay it's true for this case, this case, and this case so we can just assume it's solid until we find a counter example."

Could you PLEASE try better than wikipedia? Some kind of source where the author is public and not open content that can be edited by anyone?

And by the way..you haven't taken a first year logic course... amirite?... as you would say...hence you're none response to my question on that.

Stak: I ignored it because it's lame, Yes, I'm taking a seminar in Modal logic right now, in the context of Metaphysics.

[_marg]

and this post I wrote to Stak:
viewtopic.php?p=361065#p361065

Gee, I don't know Marg, maybe because it's trying to make a statement about all natural numbers.

Geez..Stak...find me a definition of inductive reasoning ..akin to the kind of reasoning Dawkins presented.. in other words in word format not mathematical ..that says inductive reasoning is about making a statement about natural numbers.

I'm amused the first source you finally bring up in Wikipedia, but let me put this into a better context for you:

Principle of Mathematical Induction (Mathematics)


Show true for n = 1
Assume true for n = k
Show true for n = k + 1
Conclusion: Statement is true for all n >= 1

The key word in step 2 is assume. You are not trying to prove it's true for n = k, you're going to accept on faith that it is, and show it's true for the next number, n = k + 1. If it later turns out that you get a contradiction, then the assumption was wrong.

You don't know what you are talking about..mathematical induction is not the same as inductive reasoning in logic.

Here's another source for you. http://www.cut-the-knot.org/induction.shtml

"Mathematical Induction (MI) is an extremely important tool in Mathematics.

First of all you should never confuse MI with Inductive Attitude in Science. The latter is just a process of establishing general principles from particular cases."

Here's another one http://www.earlham.edu/~peters/courses/logsys/math-ind.htm

"Mathematical induction" is unfortunately named, for it is unambiguously a form of deduction. However, it has certain similarities to induction which very likely inspired its name. It is like induction in that it generalizes to a whole class from a smaller sample. In fact, the sample is usually a sample of one, and the class is usually infinite. Mathematical induction is deductive, however, because the sample plus a rule about the unexamined cases actually gives us information about every member of the class. Hence the conclusion of a mathematical induction does not contain more information than was latent in the premises. Mathematical inductions therefore conclude with deductive certainty.


Do I need to go on?

That was simply a brief google and taking essentially the first two sites. Why don't you read those sites, the second one explains things pretty well and then do your own google on this.

Once again Stak inductive reasoning in logic does not use or need "FORM" to show the theory or conclusion is cogent or logically valid. This is in contrast to deductive reasoning in which validity entails the structure of the argument..if it is deductively structured or in deductive recognized form it is considered valid. There is no equivalent meaning of validity in inductive reasoning.

by the way I asked that you don't use mathematics to show inductive reasoning entails "form" in order to be valid..because I said that it doesn't necessarily map onto the world. And what we were discussing was Dawkins reasoning in his book. We weren't discussing mathematics, nor interested in a fantasy/imagined world, nor interested in modal logic. Dawkins in his argument is interested in the experiential world, interested in mapping his argument onto the experiential world. Mathematics has its own rules and axioms..in a sense it is it's own world ..self contained

http://www.mathdb.org/notes_download/elementary/algebra/ae_A2.pdf

Mathematics distinguishes itself from the other sciences in that it is built upon a set of axioms and definitions, on which all subsequent theorems rely. All theorems can be derived, or proved, using axioms and definitions, or using previously established theorems. By contrast, the theories in most other sciences, such as the Newtonian laws of motion in physics, are often built upon experitmental evidence and can never be proved to be true.[/quote]

[_MrStakhanovite]

Hey Marg, if you get what I'm saying...answer the question I posed to Tarski. Better yet, tell me in your own words, what I said to Tarski.

[_Tarski]

If X is a property of a given set (for example), I don't understand why this has to be considered deduction, because there would be more data in the conclusion, than in the conjunction of stated premises.

What do you mean by "data"?

I'm am often annoyed when people say things like there is nothing in what can be shown deductively that isn't already contained in the premises and axioms.
When we discover what follows from the axioms of subjects such as topology, real analysis and differential geometry, we are adding surprising things to the body of human human knowledge. We are in a sense discovering them.

Mathematics is more than some grand version of "joe is a bachelor therefore joe is unmarried".

We can all agree that this is really nothing.

But compare to the deductive climb from ZFC set theory to the surprising Atiyah-Singer index theorem.

Now, I think what may bother you for some reason is related to the fact that we conclude something about an infinite set of propositions when we do mathematical induction. Maybe it bothers you that we seem to only have a finite number of premises. Well, I am not sure what principle you are worried about violating but in fact in the case of mathematical induction there really is an infinite number of things that go into the premise side of the argument in some sense because one must prove deductively (often algebraically) that each of an infinite set of implications is true:

I_n: P_n impies P_{n+1) for all natural numbers n.

Note the "for all natural numbers n" which is often not explicitly stated.
When the induction succeeds we are able to prove all implications I_n at once by a principle of proving it for an arbitrary n (this in not to say that we prove it for one specific n or that we prove it for a few n). We prove it for all n by a single argument. That is, we establish an infinite list of implications. But we are not done:

After we have established that infinite list of implications and the truth of the initial P_1, then we are free to use the axiom of induction to conclude something about an infinite set. This last step is also deductive.


Now you seem to talk of how much "data" or how much "information" exists in premises as compared to the conclusions. I am not aware of a rigorous notion of information used in this sense although I understand the intuition.
I think the notion you are groping for is implicitly defined by the existence of a valid deduction in the first place: If the valid deduction exists then that would just be what we mean by there being the same (or no more) logical information in the conclusions than was in the premises. But is there something else you mean that I am not aware of?

[_MrStakhanovite]

Tarski, I’m probably doing a bad job at expressing myself here, but, I think the differences in our axioms might be what’s leading to my confusion. You’ve stated P5 like this:

If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.

(Sets aren’t mentioned in the previous axioms, is that referring to domains?)

And I stated it as such:

If X(0) is true and X(n) -> X(n+1) for all n then X is true for all n

It seems to me that your definition entails mine, but mine does not entail yours.

For example, in the definition I gave, what if we are not talking sets, but just integers? And we used symbols for operators ( + -) and say for some property P(n) is true if and only if n = SSSS0. I guess I’m seeing uncountable models, and that leaves things look asymmetrical to me.



ETA- I just want to remind you, this is totally self taught, so I'm probably doing this all wrong, but this is probably accounts for why I have trouble seeing all this as deductive.

[_Tarski]

OK, I don't want to be the arbitrator in the argument between Stak and Marg. But let's just look at this one thing:

marg wrote:
The speculation initially may be consider inductive..but that proof is deductive reasoning.

Stak: That is literally one of the dumbest things you could possibly say. All it is doing is saying " okay it's true for this case, this case, and this case so we can just assume it's solid until we find a counter example."

I think you two are talking past each other here and in other places.

I know exactly what Marg is talking about and it is not one of the dumbest things a person could say. I believe what she is talking about is exemplified by the case of the Riemann hypothesis.

The latter states that all the (non-trivial) zeros of the Riemann zeta function lie on the vertical line Re(z)=1/2 in the complex plane.
We do not have a rigorous mathematical proof of this as of yet (well maybe some recent attempts work but it is too early to tell).
Anyway, what we do have is knowledge that all known zeros do lie on that line and the number of cases we know is very large thanks to computers.
So inductive reasoning in the sense used in empirical science might lead us to believe that the RH is true but this is not considered valid in mathematics and would be inappropriate to the subject matter
.
If someone could come up with an inductive proof in the sense of "math. induction" or any other deductive proof (perhaps a proof by contradiction) then we would simply know that the RH is true in a much stronger sense than we know that all electrons are subject to quantum electrodynamics (since the latter would be due to inductive reasoning in the sense philosophers of science worry about).

So when people see how many cases already have been checked and this leads them to conjecture the truth of the RH, they are (kind of) thinking inductively in the sense of induction as understood by Hume etc. But this is not mathematics and we await a (deductive) proof (which may perhaps use the mathematical induction!).


So in short, I knew instantly what Marg meant and it was not stupid by any means.
(Gotta call it like I see it)
Perhaps what you thought she was saying would have been stupid--I don't know. Like I said, you are talking past each other.

I am not going to make a big judgement about Marg's intelligence, who wins, or to what extent she just googles things without understanding them but based on this thread alone I would say she does not deserve to be called stupid at all.
In fact, I am absolutely sure that she has much higher intelligence than my wife, my dear mother, my brother and quite a few other people I love and respect. None of them could come close coming up with what she came up with in this thread no matter where she got it.

So, what business would I have insulting her intelligence?

In fact, I now feel guilty that I implicitly went along with this Marg is so stupid thing.

Perhaps a rereading of old threads with her and Jak arguing this or that with Gad or CC about Godel or something might move my opinion in one direction or another (I remember being frustrated with her at the time) but I couldn't even come close to thinking that she is just "dumb" or something. That would be totally unfair and false.

[_MrStakhanovite]

Now you seem to talk of how much "data" or how much "information" exists in premises as compared to the conclusions. I am not aware of a rigorous notion of information used in this sense although I understand the intuition.

I'm thinking along the lines of First Order Logic and the Compactness Theorem. Proofs being finite by definition.