Computational Metaphysics

Photios

Is There a God-Shaped Hole at the Heart of Mathematics?

Here is a n interesting essay on the various proofs over the years on the existence of God.

Most of what is covered will not be new to most everyone here. What was new to me was the last bit.

...

"Gödel’s ontological proof uses mathematical logic to show that the existence of God is a necessary truth. “God” in Gödel’s proof is defined as a “Godlike object”. In order for an object to be “Godlike”, it must have every good or positive property. Also, a Godlike object has no negative properties. In the context of Gödel’s proof, an object (x) has “the Godlike property” if and only if for every property (φ), if φ is a positive property, then x has property φ. Because being “Godlike” is a positive property, it is possible that that property exists in an object (x). After mathematically defining essential or necessary properties Gödel then shows that it is necessary that there is an object x that has the Godlike property. If a Godlike object (i.e. God) has every good property, and necessary existence is a good property, then a Godlike object (i.e. God) must exist."

...

"In time, the fruit of Leibniz’s binary theory, AI, would demonstrate the consistency of Gödel’s ontological proof. One team of AI researchers concluded that “From Gödel’s premises, the computer proved: necessarily, there exists God” and also that “this God-like entity is unique, i.e. monotheism is a consequence of Gödel’s theory.” Such experiments, the researchers declared, “strikingly demonstrate the potential benefits of a computational metaphysics…in which humans and computer programs join forces in order to settle philosophical disputes,” thus fulfilling “Leibniz’s vision known as ‘Calculemus!’” Yet, the results of such AI calculations about metaphysics still rely on fundamental assumptions regarding the mathematical axioms that one assumes in the first place. Thus, the only weak point in Gödel’s ontological proof for the existence of God would appear to be Gödel’s own incompleteness theorems proving the limited and unprovable nature of all mathematical endeavors. But this, of course, would come as no surprise to Gödel."

The word 'concluded' in bold is a link (in the article) of the proof, found here:

https://arxiv.org/pdf/2001.04701.pdf

Thoughts about the proof? A pretty neat result, IMHO.

https://www.templeton.org/news/is-there-a-god-shaped-hole-at-the-heart-of-mathematics?

Banno

A5 Necessary existence(NE) is a positive property. PNE

There's the problem. Assuming there could be such a thing as necessary existence.

Two issues. First, this takes existence as a property. Second, it assumes that there could be something that exists in every possible world. But for any individual, it is trivial to suppose a possible world that does not contain that individual.

Can't see it working.

Edit: Seems to be introduced into the ultrafilter at lines 17-21 of fig. 3. But it's ugly s hell and difficult to follow.

Wayfarer

Not directly connected but might be relevant:

Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason. — Godel and the Nature of Mathematical Truth

“I believe that the only way to make sense of mathematics is to believe that there are objective mathematical facts, and that they are discovered by mathematicians,” says James Robert Brown, a philosopher of science recently retired from the University of Toronto. “Working mathematicians overwhelmingly are Platonists. They don't always call themselves Platonists, but if you ask them relevant questions, it’s always the Platonistic answer that they give you.”

Other scholars—especially those working in other branches of science—view Platonism with skepticism. Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.

Platonism, as mathematician Brian Davies has put it, “has more in common with mystical religions than it does with modern science.” The fear is that if mathematicians give Plato an inch, he’ll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all?

Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonism—but has since come to see it as problematic. If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?) — What is Math?

jgill

Gödel’s own incompleteness theorems proving the limited and unprovable nature of all mathematical endeavors — Photios

Hmm :chin:

“If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” — What is Math?

What nonsense.

sime

As argued in other threads, logical necessity isn't the same thing as empirical necessity or epistemic certainty. Modal logic is in general a fallacy if it's modal operators are interpreted that way, for one cannot know the properties of every possible world until one has literally counted and inspected all of them.

On the other hand, its a mistake to think of ontological arguments as being empirical arguments, for they are really an expression of faith, i.e. of deontic necessity. So the above argument is really a computational way of expressing religious faith.

Personally, I think ontological arguments are interesting when properly considered, and wonder if they have potential application in the secular religion called psychotherapy.

Agent Smith

Does existence, at all times, make something greater? For instance, taking a page out of antinatalism, a case can be made that nonexistence is (far, far) better than existence. Then there's the intriguing expression "too good to be true", suggesting the impossibility of perfection, at least when it comes to goodness, and God is good, actually omnibenevolent and thus that much more unreal.

Gödel's proof too, if I'm correct, relies on the greatness of existence. I remember meeting this woman and thinking "you're too good for me." Is the universe, this world, too bad for a being so good as God?

Signing off...

Wayfarer

“If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.”
— What is Math?

What nonsense. — jgill

:clap:

Metaphysician Undercover

Yet, the results of such AI calculations about metaphysics still rely on fundamental assumptions regarding the mathematical axioms that one assumes in the first place. — Photios

Begging the question.

javi2541997

“If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.”
— What is Math?

What nonsense. — jgill

Interesting quote indeed. I going to defend my opinion of what could be the meaning.
Empiricism is the one of the principles of philosophy where it is argued that our knowledge (adding all the stuff connected to) is based in our experience of language and interpretation. So, we develop ideas and theories staring in a basic primise: we are not born knowing, we develop our knowledge through the time making the effort of learning from others.
Then, putting the math example, we develop all the equations and theories because we are taught previously.

I guess, in Platonism is different. It is a classic thought where what is around us is full of ideas. Then, it is not necessary to work on it through experience or practice.

If maths are proven by showing the effectiveness, then it is correlated to empiricism. They both need to be together. Platonism could be just ideas.

javi2541997

Is the universe, this world, too bad for a being so good as God? — Agent Smith

This is a poor view of human values and knowledge and it looks like we are forced to be a vassal of God. I think we should rephrase your question:
is this universe, this world, worthy to be a good/honorable citizen to live in?

Agent Smith

is this universe, this world, worthy to be a good/honorable citizen to live in? — javi2541997

:confused:

jgill

If maths are proven by showing the effectiveness, then it is correlated to empiricism — javi2541997

Within mathematics proofs are the product of reasoning. However, in physics a mathematical process may show effectiveness without a reasoned proof. Feynman did this with his path integral. When this happens mathematicians try to catch up and provide a foundation by rigorous proof. Sometimes they do and sometimes they don't. There are anomalies in regularization, for example.

Banno

Seems to me a general rule of thumb is that since there is nothing in the conclusion of a valid argument that is not implicit in the assumptions, any valid argument cannot bring something into existence that was not there implicitly in the assumptions.

Hence any variation of the ontological proof must be suspect, since by their nature they seek to demonstrate the existence of something not found in their assumptions.

god must be atheist

It is true that any variation of the ontological proof must be suspect, since by their nature they seek to demonstrate the existence of something not found in their assumptions.

And if the thing they attempt to prove exists is in the assumptions, then it's circular reasoning.

Nobody can win. The game has been rigged.

-----------

I am not sure where I made a logic error in the above. If I made an error in the first place. If I was right, then it seems nothing in existence or out, has the capacity to be proven to exist.

Maybe this is the crux to the improbability of existence? If only I can be proven to exist, and my existence can be proven only to myself, then everything else is suspect for their existence.

And that is actually the backbone of the reasoning which validates philosophies such as solipsism, or skepticism, which deny the existence of the material world.

sime

Hence any variation of the ontological proof must be suspect, since by their nature they seek to demonstrate the existence of something not found in their assumptions. — Banno

Anselm's ontological argument is essentially an inductive definition of god, analogous to the inductive definition of the natural numbers in type theory.

1. Posit an initial imagined god ; g(0).
2. Given any imagined god, specify the existence of an improved 'realer' god; g(s+1) = improve g(s)
3. Define a 'perfect' god in terms of the fixed-point g(inf) = improve g(inf)

Construed this way, the god specified in step 3 isn't a deduction relative to 1 and 2, rather it constitutes the definition of a fixed point for improve with respect to the premises 1 and 2.

As with the creation of the set of natural numbers, such arguments aren't empirically meaningful so they must be imperatives, about how to think about 'god' in the case of religion, and about how to use the sign Nat in the case of mathematics, i.e. as a sign signifying an unspecified number of iterations of step 2.

Agent Smith

Nobody can win. ~~The game has been rigged.~~ — god must be atheist

The house (always) wins! Re: Pascal's wager.

Banno

↪sime

Ok, so god as something a greater than which cannot be conceived. Fine.

Agent Smith

Ok, so god as something a greater than which cannot be conceived. Fine. — Banno

God: That than which nothing greater can be conceived.

TonesInDeepFreeze

https://www.templeton.org/news/is-there-a-god-shaped-hole-at-the-heart-of-mathematics? — Photios

That article egregiously misstates Godel.

"Godel was famous for proving mathematically that all mathematical systems are incomplete"

No, Godel proved that some systems (or a certain kind) are incomplete. It is well known that there other systems that are complete.

"Godel also believed that the universe would be incomplete, and inconsistent."

Where is Godel supposed to have said said? It doesn't even make sense. What are complete and consistent, or not, are formal systems, of which "the universe" is not one. Also, If a system is inconsistent then it is complete. That is well known to any first semester student in mathematical logic.

"Godel’s ontological proof uses mathematical logic to show that the existence of God is a necessary truth."

The proof uses not just ordinary methods in mathematical logic but also modal logic with certain assumptions.

"Godel’s own incompleteness theorems proving the limited and unprovable nature of all mathematical endeavors."

Endeavors are not provable or not. Rather, formulas are provable or not relative to a system. And there is no formula such that for all systems the formula is not provable.

"1. finitely specified, 2. large enough to include arithmetic and 3. consistent, then it is incomplete."

No, It should be: 1. recursively axiomatizable, 2. expresses enough arithmetic to formalize the Godel sentence (or sometimes said as "expresses a certain amount of arithmetic" or "is an extension of Robinson arithmetic", et. al), 3. is omega-consistent (later generalized to 'consistent' by Rosser).

TonesInDeepFreeze

the inductive definition of the natural numbers in type theory.

1. [...] g(0).
2. [...] g(s+1) = improve g(s)
3. Define a [...] fixed-point g(inf) = improve g(inf) — sime

I've never seen that in type theory or elsewhere. it seems to make no sense. Please say where you have ever seen that as type theory?

* There's no mention of anything type theoretic there.

* It's not a definition (inductive or otherwise) of 'is a natural number' nor a definition of 'the set of natural numbers'.

* 'g(0)' needs to be followed by '= T' for some term T.

* 'inf' in a context such as this does not refer to a mathematical object. Multiple times in other threads, I told about your misunderstanding related to this.

/

This does make sense (transfiinte recursion), though it's not a definition of 'is a natural number' or 'the set of natural numbers):

g(0) = c
g(s+1) = g(s)
g(k) = F(g restricted to k) for F a class operation and k a limit ordinal

For a definition of 'is a natural numbers' we don't need induction or recursion. We only need:

n is a natural number iff n is a finite ordinal

For a definition of the set of natural numbers' in set theory we have:

D is inductive iff (0 is in D and for all x, if x is in D then xu{x} is in D)

the set of natural numbers = the least inductive set

sime

I've never seen that in type theory or elsewhere. it seems to make no sense. Please say where you have ever seen that as type theory? — TonesInDeepFreeze

To put it categorically, I'm referring to the definition of the set Nat as the carrier of an Initial Algebra

TonesInDeepFreeze

↪sime

So I guess you mean type theory couched in category theory. I don't know enough about that to evaluate your claims about it, but there are some general points to mention.

First, what textbooks or lecture notes form the basis of your notions of category theory and type theory, especially explaining their relationship? I would like to be able to see a systematic treatment so that I can see the full context rather than a standalone, unsupervised article in a pot luck encyclopedia.

Category theory can also be couched as a non-conservative extension of ZFC (viz. ZFC+Grothendiek_universe). So it makes no sense to denounce ZFC as you do while using a theory that presupposes ZFC. And various of your comments (in another thread) about ZFC show that you don't understand ZFC, nor intuitionism, so I have little reason to believe that you understand the even more complicated subjects of category theory and type theory.

In other threads, and now in this one, you use 'inf' as if it stands for something. But now you switch to 'N' for the set of natural numbers. 'N' is not problematic, since it does refer to a particular object, viz. the set of natural numbers. But I don't know of any texts that refer to infinity itself as a mathematical object (other than 'infinity' as a point on the extended real line and things like that, and the notation 'to inf', which resolves without 'inf' as a noun). If you would cite such texts then I might be able to look them up.

Gnomon

Gödel’s ontological proof uses mathematical logic to show that the existence of God is a necessary truth. “God” in Gödel’s proof is defined as a “Godlike object”. In order for an object to be “Godlike”, it must have every good or positive property. Also, a Godlike object has no negative properties. — Photios

I'm not a mathematician, so I don't use "mathematical logic" to prove the existence of a Necessary Being. So, while I agree with Goedel's general conclusion, my "verbal logic" indicates that a "godlike object" must be Holistic, hence encompassing all aspects of the real world : both positive & negative; both matter & antimatter; both good & evil. Even Christianity acknowledged that logic by including an evil lesser god (Satan) to blame for all the not-so-good features of the creation. In Hinduism, there are good and evil gods, but they are all subsumed under the universal unitary deity Brahman, not to be confused with the triumvirate personality Brahma.

In math, necessity is equivalent to unity, so that 1 = 1. Therefore, if there are parts of a system, there must be a whole system to unify them. If there is more than one thing, there must be a category of all things. You can't have Something, without acknowledging the necessary existence of Everything. But those math/logic abstractions are far from the conventional understanding of God. The Whole or ALL may be perfect, but if there are imperfections in the creation, the potential for negative must exist in the creator. That doesn't mean that G*D is Evil, but that in the whole system positive & negative cancel each other out : like matter + antimatter = zero matter (annihilation). But, since Energy is neither created nor destroyed, the neutral positive-negative Potential remains in the system, like a storage battery. In any case, neutral Potential energy is necessary for positive-negative Actual energy.

I'm sure that rambling reply sounds neither logical nor mathematical. But to me, Goedel's "all good, no negative" conclusion in bold, sounds more like Judeo-Christian-Islamic theology than mathematical computation. :nerd:

Necessary Being :
Spinoza and Leibniz held that what makes God necessary explains his very existence.
https://www.rep.routledge.com/articles/thematic/necessary-being/v-1
.
Mathematical Necessity :
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements.
https://en.wikipedia.org/wiki/Necessity_and_sufficiency

TonesInDeepFreeze

↪Gnomon

Godel uses modal logic and certain modal assumptions. His argument is not "mathematical computation"

litewave

In order for an object to be “Godlike”, it must have every good or positive property. — Photios

What is a "good" property? Is positive electrical charge a "good" property?

Constance

What is a "good" property? Is positive electrical charge a "good" property? — litewave

Finally, someone put the proverbial finger on this. Arguments are only as good as their definitions. You can't go on about the GOOD unless you have in place a defensible df. Here, it is God's goodness. But wait, do you have something in mind here for God's df?

The question begging is awful, like swirling clouds confusion. Talk about God has to get back to basics, conditions logically prior to fancy proofs.

Gnomon

Godel uses modal logic and certain modal assumptions. His argument is not "mathematical computation" — TonesInDeepFreeze

OK. I didn't know that. But I was responding to the OP, which mentioned "computational metaphysics". I suppose the difference between Modal Logic and Mathematical Logic is primarily in the vagueness of modal terms, such as "Necessity". If so, then I guess my own reasoning was more like Modal Logic than Mathematical Computation. Which would explain how rational people could arrive at different conclusions from the same premise. Anyway, it's not a big deal for me. The God concept will remain, as always, a debatable metaphysical opinion instead of an absolute mathematical certainty. :smile:

TonesInDeepFreeze

↪Gnomon

Mathematical logic and modal logic are related and interact sometimes. But Godel's argument about God is not mathematical in the sense that the modal assumptions would not ordinarily be thought of as mathematical assumptions.

the vagueness of modal terms, such as "Necessity" — Gnomon

The modal operator 'Necessary' is primitive, i.e. undefined.

Computational Metaphysics

Welcome to The Philosophy Forum!

Categories

More Discussions