The Lies of Richard Dawkins, Episode 6: Saint Thomas Aquinas

Great Debates
461

I’m not sure I’m clear on what you’re trying to say by ‘the relevant language of statements.’ Are you still trying to talk about things outside the system or not?

You can always add new symbols to a system to define a new system. You can always add axioms. You can always add new operators. You can even always add new logical language. Calling something incomplete for this reason, as you now claim you were always doing, is meaningless. It can be said of all mathematics. It describes everything and explains nothing. Nothing can be learned by the statement, “such and such is incomplete,” from that definition. In fact, GIT doesn’t say anything from that definition. It would be meaningless to say, “a system can’t be both consistent and complete,” if it’s not possible for it to be complete. And it’s not used that way. As you yourself admitted, Presburger arithmatic, and other systems, are taken to be complete. GIT is actually taken to mean… something. We’re just debating what.

[QUOTE=Indistinguishable]
I claim that one statement in the relevant language is “There exists an x such that x * x = 1 + 1”, and that the above axiom system neither proves nor disproves this. Thus, it is incomplete, according to the mathematical definition of incomplete.

Do you object to this? If so, what is it that you object to?
A) The list of axioms above constitutes an axiom system, for the indicated language of statements
B) The mathematical definition of “incomplete” is “Either proves or disproves every statement [in the relevant language of statements]”
C) “There exists an x such that x * x = 1 + 1” is in the relevant language of statements
D) One cannot prove “There exists an x such that x * x = 1 + 1” from the list of axioms above
or
E) One cannot disprove "“There exists an x such that x * x = 1 + 1” from the list of axioms above
?

[/QUOTE]

Ok. Fairly clear. What I object to is statement E). It isn’t true.
let’s take y to be any symbol in the system. there is no y such that y = 1+1. Because 1+1 is undefined. 1+1 does not return a member of the set we are using because the ‘+’ operator isn’t closed.

Do you know what it means for a system to be closed under some operation? If a system is closed under addition, then the addition operator, (by convention, also described as ‘+’) always returns a member of the set when used on members of the set. The natural numbers are closed under addition, but not under subtraction. (‘3 - 7’ is not a member of the natural numbers.)

The system you described isn’t closed under addition. There is no definition of the ‘+’ operator that describes what happens if you use it on x and y when x=1 and y=1. (The ‘+’ operator here can’t be formally described as ‘addition’ because of the associated properties that come with the term, such as closure.)

So, 1+1 is undefined. 1+1 is not a member of the set, so there is no y, (member of the set,) which can equal it. E) is disproved. But, I’ve already explained this in other terms. Remember when I said that the square root of 2 is undefined for the rational numbers but defined for the real numbers? And that the question “is there an x such that x*x = 2,” is still decidable for each system? You have not made an undecidable question.

I also have a problem with proposition B) because I don’t know what you’re trying to say with it.

462

Also, I’d like to clarify some things about conventions. You can always say, “let’s take elephant to mean ‘1+1,’ or 2 to mean ‘1+1.’” And if everytime you refer to ‘elephant’ or ‘2’ you are refering only to some property of ‘1+1,’ then it is a convention. When you use preconceived notions of elephant, or 2 to try to say something about the properties of ‘1+1,’ it is no longer a convention, and you have defined an axiom of the system.

For instance, (and I know you want to forget everything that came before, because it was such junk, but I want to explain why what you said can’t be termed ‘convention,’)

[QUOTE=Indistinguishable]
Well, if we agree to call whatever is in the field “numbers”, then we’ve got numbers…
Sure it… *(2 and by implication, other numbers,) *…follows from the axioms; if you have 1 and you have +, then you have 1 + 1, which we can abbreviate as 2…
[/QUOTE]

items in italics added for clarity.

2, (and obviously, properties associated with it ,) follow from the axioms if you first accept the convention?

Ok, you start with the convention: call what’s in the field ‘numbers.’ That’s fine; if you stopped there. But then, you say “then we’ve got numbers”! which is clearly assigning the common conception of numbers to what’s in the field. It’s like saying, “1+1 = elephant, therefore, 1+1 has a trunk.”

463

[QUOTE=ch4rl3s]
The interesting thing about Goedel’s result was that a sufficiently powerful system can’t be completed without becoming inconsistent.
[/QUOTE]

Apparently not. From the definition he eventually gave, it’s impossible to complete anything, since you can ALWAYS add new symbols, axioms, or operators.

464

I almost used the example of fire, myself. But fire doesn’t actually seek fuel. It either finds it or it doesn’t.

[QUOTE=Half Man Half Wit]
Anyway, what this is really aimed at – or was, originally – is your assertion that Genghiz Khan was a ‘good atheist’ because he had a lot of offspring, since all the atheist life can have in the way of purpose is reproduction (never mind even the issue with the belief in god(s) or somesuch). If you’re not still claiming that, I guess I can live with the remaining differences between the two of us.
[/QUOTE]

I had been moving away from that claim as I wanted to explore your reasoning. It looked like there may have been something to it.

[QUOTE=Half Man Half Wit]
If I get you right here, you’re essentially saying that, since everything ever since the first molecule learned the reproduction trick is just part of a mindless chain of cause and effect, then so is the decision to give your life a purpose. Well, I can’t disagree with that, but does this really invalidate that purpose?
[/QUOTE]

I can’t say that being the end result of a mindless chain of cause and effect invalidates the purpose something comes to possess.

But, since I used this argument to describe the purpose of life… You have inadvertently validated my original claim. I’m going to have no choice but to go back to accepting it.

[QUOTE=ch4rl3s]
… Life did not form for any purpose. Absolutely incidental.

But, once life comes into being, it exists for the purpose of reproducing itself. It acts to do this. It seeks fuel to continue. It attempts to continue… This is the only natural purpose in the universe…
[/QUOTE]

except here.

You ranked ‘function of life’ below ‘predetermined goal orientated purpose,’ as well as below ‘meaning of life.’ I don’t know how you intended ‘predetermined goal oriented purpose’ to rank next to ‘meaning of life.’

What was that then?

465

Same as the cell. It doesn’t any more form the thought ‘boy, I sure could gobble up some juicy amino acids right about now!’ than the fire has any wish to seek out some nice carbon to oxidise.

You keep confusing yourself, both regarding my and Indistinguishable’s arguments. I haven’t said anything about rank – my usage of the word ‘higher’ was pretty obviously a figure of speech often used in this context, more akin in meaning to ‘preordained’, or even, to point directly to what we’ve been sneaking around all this time, ‘god-given’ --, I have argued that purpose has two distinct meanings, pre-determined goal and function, with pre-determined goal in this case meaning that something is set up with the explicit intent to fulfil a certain task, and function here meaning a result incidentally achieved by something – i.e. in other words, the pre-determined goal can in a way be taken as the reason for the existence of something, while the function is merely a chance by-product of something’s existence (call them ‘a priori’ and ‘a posteriori’ purpose, respectively) --, and that you’re equivocating between the two.

Your inference that I consider one to be ‘below’ the other is entirely your own fabrication, particularly since I explicitly stated that I don’t see any way to compare both. My own argument merely is that life accomplishes the function of reproduction incidentally to its existence, yet that pre-determined goals can be created, for instance if somebody builds a machine to lift a load, or some other contraption like that, and, in a similar way, such a pre-determined goal can be set for someone’s life (by that someone).

Hence, life has an a posteriori purpose in reproduction, but not an a priori one; however, the individual can give (some portion of) his life an a priori purpose by, well, deciding on one. Your whole argument amounts to – fallaciously, in my opinion – equating both meanings of the term in order to obtain a means of comparing the relative merits of ‘atheistic’ lives, which doesn’t appear to make a lot of sense since it seems that such a comparison is only possible for contraptions in regard to their ability to fulfil a priori purposes – something can’t be ‘better’ at achieving something if it never set out to (or was set up to) achieve it in the first place. Hence, the fact that reproduction is only incidental to life invalidates your reasoning as to Genghiz Khan being ‘the best atheist’ (and that it ever landed you in such a ridiculous place should have been a clue, frankly).

And as for your accusations of dishonesty towards Indistinguishable, by now it appears like you deliberately choose to read his words in a distorted way – everything he’s said so far, best as I can tell, can be understood as being perfectly well in line with common mathematical usage, and it requires some impressive contortions you seem all too willing to perform to read any sort of sly duplicity into it; and even if he sometimes was less than strict, terminologically, his points are reasonably clear, and valid. In fact, it seems your argument has reduced to mere semantical quibbles as opposed to a discussion of actual content, so perhaps it would be appropriate to get back to the origin of this whole argument, your assertion that

which is a bit overreaching in stating the philosophical implications of Gödel’s result – it’s true that there are those that argue that the incompleteness theorem makes a theory of everything impossible, most prominently Stephen Hawking, but even this doesn’t say anything about a need for ‘extra-universal’ explanations (it merely makes physics inexhaustible and ensures that there’ll always be a job for theoreticians), and it’s also been noted that this depends on your notion of what, exactly, a theory of everything ought to give you – a complete description of all phenomena in the universe, i.e. the ability to expressly calculate the universe’s state in the future from the knowledge of its state now (or in the past) would probably run into problems with Gödel, if perhaps only conceptual ones (it might well be that all physically interesting statements are perfectly well decidable), despite of course flying in the face of the current understanding of non-determinism in quantum mechanics and chaotic behaviour, but a simple statement of rules underlying all dynamics isn’t much fazed by the fearsome Gödelian spectre.

466

We’ll see what I can get to today, but let’s start with this.

Did I ever do that? I was trying very hard not to. I recognised that it was very likely such errors could be made in ignorance, but I will admit that it crossed my mind he might be doing it on purpose.

The closest I can find to saying anything like that is this:

[QUOTE=ch4rl3s]
Starting by claiming it is in the system and now claiming you’ve only been using it as a convention, (as well as being untrue,) is truly, truly, bad form.
[/QUOTE]

I was simply making a statement of fact, without making any judgement as to whether anyone was bing dishonest. If you claim a symbol is a member of a set, then claim you were not claiming it was a member of the set, (only using it as a shorthand convention to describe another set of symbols that were members of the set,) that would be an untruth. Saying the statement is untrue. Not saying he didn’t convince himself he was only using it as a convention.

[QUOTE=Half Man Half Wit]
– everything he’s said so far, best as I can tell, can be understood as being perfectly well in line with common mathematical usage, and it requires some impressive contortions you seem all too willing to perform to read any sort of sly duplicity into it…
[/QUOTE]

Did you even look at what can actually be demonstrated from what we’ve said?

[QUOTE=Half Man Half Wit]
…and even if he sometimes was less than strict, terminologically, his points are reasonably clear, and valid…
[/QUOTE]

No! They aren’t!
Let’s look at what can be verifiably demonstrated!
Mathematicians believe that Presburger Arithmetic, and other similar systems, are complete. They believe that more powerful (and/or expressive) systems can be made complete, but not without making them inconsistent. Presburger Arithmetic can be formed by removing axioms from a more expressive arithmetic, (Peano Arithmetic.) Godel stated that you **could not complete **a formal system without making it inconsistent. These are all demonstratable facts. I’m going to add the ‘personal’ claim, (as if I’m the only one who would claim this,) that Godel meant that formal systems could be completed, (I’m not going to claim this as demonstratable.)

This is what Indistinguishable has claimed he was saying about completeness all along:

[QUOTE=Indistinguishable]
Removing axioms from an axiom system will only cause it to be able to prove less. So, what I was saying was, if you start with an incomplete axiom system, and remove axioms (while keeping the language of statements one is dealing with the same), then the system one ends up with will necessarily be incomplete as well [it can’t prove anything the starting system didn’t].

You make oblique reference to the completeness of “impoverished” systems such as Presburger Arithmetic and so forth. But this doesn’t contradict what I was saying. Sure, you can think of Presburger Arithmetic as Peano Arithmetic with all the axioms about multiplication ripped out… and if you think about it that way, then Presburger Arithmetic is incomplete
[/QUOTE]

He has also claimed that is the ‘mathematic’ definition of complete. Let’s assume that Presburger is incomplete because it says nothing about multiplication, (without ever explaining why multiplication is necessary,) then I can demonstrate that even if we take Peano, which includes muliplication, and add axioms to make it say everything possible about multiplication, it still can never be complete because it says nothing about an operator I’m about to invent. I’m going to call it Bazoongas. (when I told my room mate that, he said, “oh, you absolutely have to do that. Math needs more Bazoongas.”) I’m not yet going to describe it’s actions on the members of the set. You realize that addition was invented, that multiplication was invented, that we can always invent new operators; that we can always add new members to whatever set we start with; that we can always extend the logic we use… (you realize that propositional logic was invented; that it was extended by the invention of first order logic; which was extended by the invention of second order logic; there are more extensions and we can always create more.)

And do you realize from this that he’s claiming that nothing in math can ever be complete? But, but, but… (whimper…) how on earth am I going to reconcile that with the demonstratable fact that mathematicians believe many systems are complete or can be completed??? (how???) Do I dismiss what is demonstratable fact? Or do I dismiss Indistinguishable’s definition of what is ‘mathematic completeness?’ But Half Man Half Wit has claimed Indistinguishable is most often right about these things!!! Don’t I have to believe him? Whatever am I to do??? Well, why don’t you guess what I’m going to do?

Now. Let’s look at Godel’s claim that no sufficiently expressive formal system can be completed without becoming inconsistant. Indistinguishable claims it has no serious implications. I can see that, from what he believes ‘complete’ means. No formal system can ever be completed, so, you never have to be concerned with completeness causing inconsistency, so, of course, this would have no consequences. There would never be a reason to even state this. Apparently, Goedel was a fool. Or Indistinguishable is wrong about what is a mathematic definition. My claim was that Godel believed that formal systems could be completed. Whose claim makes more sense?

[QUOTE=Half Man Half Wit]
In fact, it seems your argument has reduced to mere semantical quibbles as opposed to a discussion of actual content, so perhaps it would be appropriate to get back to the origin of this whole argument, your assertion that
[/QUOTE]

[QUOTE=ch4rl3s]
the implications of Godel’s proof are that to have a theory that can explain every interaction in this universe, you have to start with a system that involves things outside the universe.
[/QUOTE]

Sigh. Do you realize how we got to this intricate math discussion?
I claimed the statement above; which is a proposition about how to derive a ‘theory of everything,’ (I meant everything in this universe, and without it becoming inconsistent. I hope that implication was clear.) And it started with a premise that Godel’s theorem had those implications.

With me so far? I have a proposition based on a premise, right? Certain people have **not accepted **my proposition because they **don’t believe **the premise! If I am to **establish the proposition **as valid to those people, I must first establish the premise!!!

Please, please, please, answer this question: Can I establish a proposition, (based on a premise,) without first getting people to accept the premise??? Can I? Please answer this one!

But you are asking me to stop trying to establish the premise, and get back to proving the proposition. That is truly, truly, truly bad form.

And also, please answer this question: Are you actually asking me to stop trying to establish the premise and get back to proving the proposition? Would you truly ask that?

[QUOTE=Half Man Half Wit]
which is a bit overreaching in stating the philosophical implications of Gödel’s result – it’s true that there are those that argue that the incompleteness theorem makes a theory of everything impossible, most prominently Stephen Hawking, …
[/QUOTE]

I can explain a little more about that claim in a moment. And I purposely decided not to use this example of a physicist who believed these implications, as prominent as he is, because I didn’t think a single example would hold any weight with this particular audience. Do you personally find the single example of Indistinguishable more believable than the single example of Stephen Hawking? Or did you accept my premise in the first place and were supporting Indistinguishable’s attack on it for some other reason? If you **accept the premise **I **can actually discuss the proposition **with you.

[QUOTE=Half Man Half Wit]
but even this doesn’t say anything about a need for ‘extra-universal’ explanations (it merely makes physics inexhaustible and ensures that there’ll always be a job for theoreticians), …
[/QUOTE]

Let me give an example, (which will try to address some of your other concerns about what we may actually need for a theory of everything, as well.)
Let’s start by assuming, (just for the sake of argument,) that the universe could be described by a completed formal system consisting of the natural numbers, a set of axioms, and addition and multiplication. Let’s further assume that this complete system falls under Godel’s theorem; it is inconsistant. To describe the universe, we need it to say everything true about the natural numbers under addition and multiplication with these axioms.

We could simply never complete it. It will be consistent, and only describe things in the universe, but it will never thouroughly describe the universe.

But, what if we could start with a consistent and incomplete system of natural numbers, with addition and most of the implications of multiplication, (we have these now,) and extend it with new numbers, say the rationals, and/or a new operator, which has the implications of multiplication that our incomplete system lacks. This new system may say everything we need it to say about the natural numbers, addition, and multiplication, but it won’t be complete itself, because it won’t say everything about our new numbers or our new operator, and it could still be made to retain consistency. But it will now assert some things about the new numbers and operator as true. But these things would not be things in the universe we know, and may never be observable. This would be a theory that starts with a model larger than our observable universe in order to encompass every true thing about our universe, while still being a consistent and incomplete system.

But is it realistic that a system encompassing only the universe would need to be complete? I think it probably is. I think a theory would have to explain all the particles in the universe, as well as describe space and time, and every possible way those particles interact with each other and how they effect spacetime around them. And that’s without proposing that it describe where, and in what state, every particle ever was for all time.

But, what if I’m wrong and it is just possible that our universe only needed some of the implications of multiplication? (going back to our simplistic model for illustrative purposes.) Well, we may just be able to eventually find a system that has just those and only those implications without being complete and remaining consistent. Or, it would still probably be easier to find a system with most of the implications of multiplication, and recognize that some of the true statements in this system describe things that are not in our universe. (i.e. a theory, like I stated before, that is larger than needed to make sure we encompass every true thing about our universe.)

oops. running out of time, gotta go.

467

No need to go so hyper, ch4rl3s. The matter is a rather simple one: say you’ve got a formal language L, and a formal system S specified in L by a set of axioms A. S is complete if every sentence expressible in L or its negation can be deduced from the axioms. Now, let A be the axioms of Peano arithmetic. From those, you can go to a set A’ of axioms that encapsulate Presburger arithmetic, by ‘ripping out’ those concerning multiplication. Done this way, Presburger arithmetic is indeed incomplete, because it cannot decide all sentences of L – in particular, those involving multiplication are omitted entirely. That’s why, through removing axioms from a system, you can only get greater incompleteness.

However, you are very right in stating that Presburger arithmetic is generally called complete. But that’s just because Presburger and Peano arithmetic do not use the same language! The axioms of Presburger arithmetic are sufficient to decide all sentences in its language, but not in that of Peano arithmetic (obviously). If you thus simply reduce the axioms, without appropriately constraining the language, you will always get an incomplete system.

That’s precisely what Indistinguishable said:

I don’t really see how you can interpret this differently.

As for the relevance of Gödel’s theorem to a theory of everything, as I said, it depends greatly on what you want to call a theory of everything – the way I see it, it could well be a statement (or a set thereof) in some mathematical framework that encapsulates the dynamics of some object of fundamental ontology – some sort of string, or brane, or something else. These then give rise to spacetime and the elementary particles complete with their dynamics, i.e. physics and nature as we know it. Taken this way, a ToE has nothing to fear from Gödel; and indeed, this seems to be the most common notion (and I was somewhat surprised to learn that apparently, differing notions exist at all).

Best as I can tell, those claiming a relevance of Gödel’s result towards a ToE seem to equate that theory with the whole of the mathematical framework its statements are formulated in – something I find rather puzzling, truth be told. It seems to me rather like claiming that you can’t formulate a theory of, for instance, the dynamics of a thrown baseball because Peano arithmetic is incomplete.

However, even if one takes the ToE to be equivalent to its whole mathematical framework, it isn’t a given that there are actually any physically relevant statements of that theory that are independent of it. Think of the universe as analogous to the baseball: its trajectory, and all of its possible trajectories are perfectly well expressible, and all its parameters decidable, within mathematical formalism, even though that formalism itself may suffer from Gödelian incompleteness.

468

I whole heartedly accept your analogy of the universe as a baseball including all its possible trajectories and parameters. But, I’m not going to discuss it right now for the following reasons…

  1. You have made it clear that you still don’t accept the premise of the proposition. I had started discussing the proposition again under the assumption that you were accepting the premise.
  2. I really do want you to answer this question:

I’m going to insist, because I need to know if we are operating under the same logic system. And then: Did you actually ask me to prove the proposition before I had established the premise to your satisfaction? Did you, in fact, make a mistake?

An honest debater can admit when they’ve made a mistake. That being said, I’m going to have to admit to making 2 mistakes over the last several posts…

  1. I totally misread the axiom system Indistinguishable listed in post #452. I still don’t know how. I just went back earlier and realized that I had formed the matrix of solutions wrong. I may have carried over properties from an axiom system I’m more familiar with, (i.e. the field axioms we had been discussing previous. I still contend that Indistinguishable did this, and still hasn’t admitted it.) or I may have misread a ‘+’ axiom as a ‘*’ axiom and vis-versa. Whatever, I got it wrong. ‘1 + 1’ is defined in the system. 1 + 1 = 0. It’s ‘0 * 0’ that is undefined.

It doesn’t change the result, though.

Let’s look at what we can prove:
1 + 1 = 0;
when x = 1, x * x = 1;
when x = 0, x * x is undefined; ‘undefined’ is not a member of the set.
does x * x = 1 + 1 in either case? NO!!! 1 != 0; undefined != 0. (!= means ‘does not equal.’ I expect Half Man and Indi to know that. just clarifying for other readers.)
The statement is disproven. Proposition E) is false.

I know he was trying to produce an undecidable statement, but he’s failed.

  1. My other mistake…

Ah… L = {first order logic (with equality), 0, S, +, *, etc.} where S is the successor function.

I’m sorry. I just found this formalism today. I had assumed that the language was the logical language, and listing the operators and explicit members of the set was a convention and that the set and operators were defined by the axioms… I mean, they are defined by whatever axioms you use, but, I totally missed what you were saying. And it’s perfectly valid.

But, let’s see if it changes the result. Peano’s axioms:

  1. 0 is an element of our set, N; (0 is a natural number)

  2. For each x in N, there exists exactly one S(x) in N, called the successor of x

  3. S(x) does not equal 0. (0 is not the successor of any number)

  4. x = y if and only if S(x) = S(y)

  5. If 0 is an element of the set, K; and for every x in K, S(x) is in K, then K = N. (the axiom of induction and the only second order axiom.)
    These define the set, N that we will be using, and the workings of the successor function.

  6. x + 1 = S(x), for all x in N.

  7. x + S(y) = S(x + y), for all x, y in N.
    These define the ‘+’ operator.

  8. x * 1 = x, for all x in N.

  9. x * S(y) = x * y + x, for all x in N.
    These define the ‘*’ operator.

(‘+’ and ‘’ can properly be called addition and multiplication in this system.)
Sorry, once again. I assumed that removing axioms 8 and 9 would remove the language to ask questions about multiplication. So, let’s remove those axioms, leaving the '’ operator in the language and see what happens. Is the system complete or not? Can it ‘prove or disprove any statement’ that can be formed from the language?

[QUOTE=Half Man Half Wit]
Now, let A be the axioms of Peano arithmetic. From those, you can go to a set A’ of axioms that encapsulate Presburger arithmetic, by ‘ripping out’ those concerning multiplication. Done this way, Presburger arithmetic is indeed incomplete, because it cannot decide all sentences of L – in particular, those involving multiplication are omitted entirely. That’s why, through removing axioms from a system, you can only get greater incompleteness.
[/QUOTE]

Let’s view a few statements.
x * y is in N, for some x, y in N. -> well, this statement is false
x * y is not in N, for all x, y in N. -> this is the negation of the above, and it’s true.

z + x * y is not in N, for all x, y, z in N -> this is also true, showing that addition is not defined to act on symbols outside the set, N.

This ‘variant’ of Presburger still tells us everything that the system can about the ‘*’ operator as defined. (or not defined as the case may be.)
In fact, for every statement that includes a term of the form ‘x * y’ we know that this term is undefined, and is not an element of N. We can replace it with ‘a,’ where ‘a’ is not an element of N. We know that addition and multiplication on undefined terms are undefined. And we can make a decision about the truth of the statement based on this. e.g.

x + y * z = w, for all x,y,z,w in N; is a relevant statement. I’m sure that Indi would have said it’s undecidable, but it’s perfectly decidable… ‘y * z’ is undefined, so
x + a = w. but ‘x + a’ is undefined, so
a = w. but, a is not a member of N, while w is in N.
So, it’s false.

I’m going to submit as a proposition that every system with an ‘element’ in it’s language, that does not appear in it’s axioms functionally reduces to a system without that ‘element’ in its language. (let’s make that more strict and say ‘an operator or set member,’ in place of ‘element’ above.)

And I’m going to say that it’s more obvious than this: (edit, ‘this’ being the following quote from Half Man Half Wit.)

I would like to get back to the baseball and it’s trajectories at some point.

469

You know, I don’t think I’m really comfortable with your debating style, so I think it’s probably best if I just back out of this and save us both some grief. Besides, this has really gotten too far afield to achieve anything useful through the continuation of the debate.

As for that one question you insist on having answered, no, there is no use in making an argument if there is disagreement regarding the premises. Hope this helps you in some way.

470

I admit I’m going to be addressing these points out of order, but not in any attempt to misrepresent what was said. I’ll leave the link so people can easily read it in context.

I really don’t understand this. I’m sure you didn’t think this decision through clearly. We’re finally, finally on the same page. If anything could ever be decided, this is the point at which it could be done. Not only that but I took a giant leap to your side for the sake of truth. I know it was a long time in coming, but I finally saw the evidence for it and swallowed my pride to admit it.

You agree that:

Yes, it does help. Because we have finally reached the same starting point for the premise. If ever there was a chance to decide something, and if ever there was a chance that I could be persuaded to your side, this must be it. I think I’ve established that I can be persuaded to the truth. I jumped whole heartedly to your starting point, and had to take quite a bit of time to reevaluate my whole position. I didn’t come to the same conclusion you did, but I admit I’ve already been drastically wrong. If you have a good argument for undecidability, I am ready to listen. But you want to stop the argument before you even attempt to give me one. Why on earth would you stop the debate just when I’m finally on your side. If there’s a good argument against what I’ve said, now is the time I should hear it. One good argument could blow my premise out of the water, which would invalidate my proposition. Why stop now?

But, this is your opportunity to change that. I’ve finally agreed that I was drastically mistaken. I can’t for the life of me understand why you would suddenly give up just when I was willing to listen.

471

Aquinas’ argument reminds me of that old Steve Martin routine.

HOW TO BE A MILLIONAIRE

First, getamilliondollars, then…
HOW TO EXPLAIN THE EXISTENCE OF THE UNIVERSE

First, youhavethisincrediblegod, then…