[2] We showed in Book I that God is neither a body nor a power in a body. But, if He assumed flesh, it follows either that He was changed into a body or that He was a power in a body after the Incarnation. It seems, then, impossible that God was incarnate.

He enumerates no less than thirteen such objections, and then deals with them in ch. 41.

http://dhspriory.org/thomas/ContraGentiles4.htm#40

et seq.

But we are only in Book I ch. 27, so it’s early days yet.

]]>Plus, there are countless number of churches (none true Christian) that have SSA ministers and proclaim the behaviour to be moral. So in actuality, the theocratic people are one of the few saying it’s not genetic or biological.

You mean the truly theocratic folks. Tell me, which human organization is the keeper of the authoritative list on what constitutes a True Christian?

[…] some countries consider murder a social value—Romans routinely murdered rulers to change governments.

Some countries still do, but call it targeted (or extrajudicial) killing or selective assassination (as opposed to indiscriminate assassination, which, well that does happen now that I think about it). But I’ll be darned if I can figure out what that’s got to do with consenting adults’ behavior in the privacy of their own bedrooms.

]]>Oops, looks like I didn’t…

]]>Yeah, we Christians have never been very good in math! ;)

]]>“Global warming, when purported science becomes religion.”

So do you think CO2 doesn’t absorb infrared radiation?

Or instead, do you think the Earth doesn’t emit it?

Let Pr(X,P) be a predicate equivalent to “Statement X is provable by Prover P”

It has the properties:

[ There exists P such that Pr(X,P) ] implies X

Not-X implies [ For all P, Not-Pr(X,P) ]

i.e. Statements provable by some prover must be true, and untrue statements are not provable by any prover.

So, “This theorem is not provable by any organic brain.” is a self-referential statement equivalent to:

S = [ Not-Pr(S,OB) ]

where OB is the collection of all organic brains, actual and potential.

The proof goes as follows:

Pr(S,OB) implies [ Pr(S,OB) ] ……………… Tautology

Pr(S,OB) implies [ S ] ……………………. ‘Provable implies true’ property of Pr

Pr(S,OB) implies [ Pr(S,OB) and S ] ……….. Combine the above two

Pr(S,OB) implies [ Pr(S,OB) and Not-Pr(S,OB) ] . Definition of S

Pr(S,OB) implies [ False ] ………………… Contradiction

Not-Pr(S,OB) or False …………………….. Definition of implication

Not-Pr(S,OB) …………………………….. Tautology

S ………………………………………. Definition of S

That part’s pretty straightforward. Now there are some more difficult questions I would expect intelligent people to ask, having got this far.

1) Does the self-reference pose a problem?

2) The predicate Pr is defined in English, is that precise enough? Does it exist?

3) You have just given a proof, even though you yourself are an organic brain. Doesn’t this contradict the conclusion?

1) A lot of paradoxes are constructed using self-reference (this proof is modeled after the Curry paradox, for example), to the point where a lot of people regard all form of self-reference as illegitimate. However, there are ways to re-write self-referential statements without self-reference (Y combinators) at the potential cost of generating an infinite tower of definitions, and there are many self-referential definitions that are useful or even essential in mathematics.

So the tricky task is to draw the line somewhere so as to include as many of the useful bits as possible while rejecting all the paradoxes. That’s hard, and different solutions have been proposed. If you reject the legitimacy of all self-reference, or of any of those tricks used to unwind it, then you’re going to reject this one. If you accept some of them, you might – I can’t say. Godel’s cleverness was in finding an indirect way to implement the self-reference via statements of arithmetic or number theory. Rejecting his approach would require rejecting arithmetic, which would be a step too far for virtually any mathematician. But that’s all it was – a way to make the self-reference palatable and to clearly define its limits.

2) To be able to talk about proving and provers, some extra formal machinery is needed. One particular variety of prover is defined by giving a set of rules and axioms, and specifying a process of sequentially exploring every possible sequence of steps looking for the statement it’s trying to prove. Exhibiting a list of such steps is sufficient to demonstrate that the prover would succeed. The prover is restricted: the sequential exploration limits it to a countable space, and often requirements of finite time and memory are imposed. (Or you could add every true statement of arithmetic as an axiom, and get a prover that will complete any proof in one step.)

But that’s not the only way you can do it. You can have provers exploring options simultaneously in quantum superposition, supertask provers, provers with specialised oracles, all sort of things. You can also have provers with tighter restrictions: bounded time and memory, for example.

‘Organic brain’ is not a precisely defined category. Does a brain with non-organic implants count? Should you allow a brain that stores most of its knowledge on the internet? However, for our purposes, all we really require is that you and me are organic brains, and there can be provers that are not organic brains. The first is not in doubt. The second I can imagine you arguing with, but given that most provers in the literature are implemented with some sort of Turing machine, I don’t see that as an obstacle.

For the predicate to be applicable to S, the prover has to be able to reason *about* provers. It has to be able to work out whether a given prover would be able to prove a given statement. It also ought to know what sort of prover it itself is.

3) And this is where you get the contradiction. The prover attempts to show that it can generate the above proof, but it also has an axiom in its database that says: “I am an organic brain.” (Or even “I might be an organic brain.” It doesn’t necessarily know.) That contradicts the assertion. The failure to check this is the illogical step I mentioned.

The prover can generate the proof by pretending to be a different sort of prover, simulating it. The prover can generate the proof by ignoring the issue, which can be interpreted either as assuming you’re the sort of prover where it doesn’t matter (which is effectively the same as above), or as being an invalid prover, capable of generating ‘proofs’ of false statements.

An organic brain asserting that it has proved S is making a false statement. It knows it is an organic brain, or might be, and is saying something that contradicts this. Proving the statement S is equivalent to saying “I am not an organic brain,” which for some provers might be true but for you and me is false. Organic brains can’t do it; not and be valid provers.

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In your strawman version, you said: “Since S says of itself it is not provably true, it therefore is provably true.” S does not say of itself that it is not provably true. It says it is not provable *by organic brains*. This is clearly a different statement.

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Out of interest, this is the same phenomenon as occurs in the ‘unexpected hanging paradox’. The judge condemns a prisoner to hang on some day during the following week, but specifies that the prisoner is not to be able to deduce the day beforehand. The prisoner uses a simple induction proof to show that this is contradictory, and that he cannot be hung on any day. His hanging therefore comes as a completely unexpected surprise to him, as required.

Again, we’re talking about proofs and deductions where the identity of the prover/deducer is one of the terms of the expression. This makes its provability indexical – it’s provable/deducible for some provers/deducers, and not others.

A lot of people look at Godel’s theorem being applied to algorithmic reasoning, and feel smug and superior that they can see the truth of the statement while the poor algorithm cannot. But it’s a general property of any prover sophisticated enough to represent a prover’s identity included as a term in the expression, as that allows statements that are prover-indexical. A universal Turing machine able to simulate such a prover certainly can. So can humans.

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