Beanybag wrote:I don't know what I believe about numbers or how I feel about the number 0. Infinity could be a real concept if Platoism is true, it could be not if phenomenalism/physicalism is true, it could be semi-real if formalism is true, and who knows if logicism is true. Intuition isn't always the best concept. Besides, am I to believe that since the world is finite, any number greater than the Universe is non-real? What then of Graham's number? Numbers don't necessarily need to map to the Universe in a one-to-one fashion, and infinity can be captured through finite computation and understanding. I don't know what to think of infinite regress arguments for this reason, however, I do defer to rejecting them for now.
However, I don't need to justify numbers in order to use them much the same as I don't need to understand a motor in order to drive a car. I am fine with, as I have been saying, sticking to a top-down approach and assuming away my foundations for now. I am not yet learned enough to provide these justifications, and I don't know if I ever will be. It is a humble acceptance that I don't know the answers.
Well, theres an easy answer to this sometimes. You say that "I don't know what I believe" or "How I feel about the number 0". I want you to take notice that you've already affirmed you exist. More on that in a second.
2nd, Theres no good reason at all to believe the Platonist can claim that abstract objects are non-mental (and non-spatial, too). Thats silly considering that Platonism already affirms that a mental substance exists. To try to say that theres some non-mental object that exists independent of you is the most unsuccessful argument I've ever heard. Thats like saying "It just does exist". Its completely unjustified, and impossible. Believe me, I tried.
If there is this non-mental objects, how would it inform you of its presence? It just did? Thats phenomenalism. Thats not rational by any stretch.
And whoa whoa whoa! Hold the presses here! We made a breakthrough! Let me quote it again
Besides, am I to believe that since the world is finite, any number greater than the Universe is non-real?
This is absolutely staggering. You understand what greater is. Most importantly, you can conceive of something greater than the Universe. You got the right idea, but its isn't a number. More on this in a second.
What then of Graham's number? Numbers don't necessarily need to map to the Universe in a one-to-one fashion, and infinity can be captured through finite computation and understanding.
Whoa whoa whoa! First off, lets keep rational about numbers.
2nd, you say that infinity can be "captured" through "finite computation and understanding". Can you clarify? More importantly, wouldn't agree that it is GREATER to have an UNDERSTANDING as opposed to appealing to a mathematical set theory you don't even know exists? Again, how would these "non-mental" objects inform you of their presence? By way of phenomenon? Surely, you must understand that there is no rational basis for such a view.
I don't know what to think of infinite regress arguments for this reason, however, I do defer to rejecting them for now.
I would also put off numbers as non-mental if I were you.
However, I don't need to justify numbers in order to use them much the same as I don't need to understand a motor in order to drive a car. I am fine with, as I have been saying, sticking to a top-down approach and assuming away my foundations for now. I am not yet learned enough to provide these justifications, and I don't know if I ever will be. It is a humble acceptance that I don't know the answers.
No, sir. Thats pretty poor considering you admitted to not having justification. That fatal flaw on top of that is that you're giving bad analogies. That "top down" approach sounds like the naturalist fallacy. Its phenomenon reasoning.
Like, I don't need to understand how a gun works in order to shoot people with it, I don't need to understand drunkenness in order to operate a car, or I don't need to understand rape in order to have sex.
While its true that you don't need to know how or why to carry out the action, you then would be admitting that you simply don't care if you're wrong. Is that the case?
Beanybag wrote:First-order logic is sound and complete for all statements of truth, however, certain paradoxes concerning knowledge create an infinite paradox for knowledge under first-order logic, i.e. it seemingly can't account for all knowledge despite its soundness and completeness.
Heres a thought. Those "certain" paradoxes you are talking about. Don't you think its possible that those paradoxes aren't valid propositions?
Beanybag wrote:You can also construct other perfect (sound and complete) axiomatic systems but they will be much more limited in power than most mathematical systems we are used to, and it would deny much of the mathematics we have established.
Key word I want to point out here is "limited". More on that in a second.
Beanybag wrote:I really enjoy set theory and I find the ZF axioms to be very fantastic, even if their completeness is demonstrably untrue (as per the continuum hypothesis and the axiom of choice), but it has been shown to be consistent up to ZF axioms without the axiom of choice (axiom of choice makes for some problems in that if it's true, ZFC is obviously inconsistent, and if it's false, ZF is obviously inconsistent - intuitively at least. What do we do???). I think it's fantastic to have such open and unsolved questions at the heart of the foundation of mathematics, it's exciting to me. However, it leaves open the problem that many of the mathematical systems we use might be shown to be inconsistent at some level.
Yeah, math is fun stuff. I don't take much interest in it anymore these days. Mostly because I've been out of school for quite some time.
Beanybag wrote:Another problem is that mathematics can't be shown to be reducible to logic (especially considering that set theory lacks completeness while first order logic does not). What if mathematics isn't reducible to logic, but they are very similar and different things? That would be extraordinary. I can't say for sure though. And that's roughly the story of my life - this omnipresent form of agnosticism.
Well, they have different purposes.
Beanybag wrote:To deny res cogitans is to assert res extensa only if I have positively denied res cogitans (which I haven't, I just haven't necessarily accepted it) and only if the law of excluded middle holds (which would assume First-order logic).
Existence falls in 2 categories. Spatial or non spatial. Since we aren't certain in the existence of res extensa. Since it could be a dream. We are forced to res cogitans. The content of the idea in the "excluded middle" is also in doubt. It could be wrong, too.
Beanybag wrote:That's another problem I have with cogito ergo sum - it hasn't provided justification for the existence of logic by which to infer a conclusion.
That because logic doesn't exist either, yet. I've spoken to another user here who has made that same mistake. The statement is not one based in logic. Its a foundational statement.
I doubt my sense experience.
I doubt mathematics.
I doubt logic. Doubt presumes there is an I. The existence of I cannot be doubted.
Beanybag wrote:But how can we prove the existence of logic without asserting our own existence?
Logic isn't needed. Ideas are indubitable as ideas.
For example, going back to the dream we could be having.
I could be having a dream that I am having fresh cup of coffee at a cafe in Seattle. I can doubt that I am having that fresh cup of coffee, but I cannot doubt that
I think I'm drinking a cup of coffee at this cafe.
Another example, I think I'm experiencing a discussion at Godandscience.org. I can doubt I am experiencing Godandscience.org, but I cannot doubt that I think I am experiencing it.
This is to say the idea is indubitable. Its is an indubitable and trivial statement.
Like, All bachelors are unmarried. Once you know what bachelor means, and once you know what unmarried means. The ideas are necessarily connected and are necessarily true. Its not a statement of logic. Like, the idea of "A" is that same as the idea of "A".
Beanybag wrote:Are we to treat our existence as a possible contradiction until logic has been proved?
No, we certainly exist.
Beanybag wrote:Or would that be too logical at this point in time. It's a little messy, as you can see. Theism certainly helps make sense of such a dilemma (which I know you're being careful in asserting, I'm not sure why), but I feel there's other solutions as well.
Well, I think its best I saved this for last. As far as what I've show. I must exist as a mental, non-spatial substance. Ideas are indubitable as ideas. Now, I have to prove God exists. There are plenty of times where you admitted to your understanding of words like "limited", "perfect", "imperfect", etc. These ideas are indubitable. I move forward in this manner. Clarification here though. A mind isn't an idea. Its a foundational reality.
I have the idea of a mind that is not my own that has ALL possible perfections. A perfect mental substance; to be more exact.
We can then ask ourselves a pretty serious question. How did we get this idea? It certainly didn't come from me. Imperfect minds can't make this up, can it? Nothing in my sense experiences (even if they are right) is perfect.
This is where I hit you with all the ontological arguments. I changed Anselm's a bit to my own liking.
You are convinced that Perfection exists in the understanding alone, at least, than which nothing perfect can exist in reality. For when you say this, you understand it. And whatever is understood, exists in the understanding.
But it is greater for Perfection to exist in reality than for it to exist in the understanding alone.
We are forced to conclude that Perfection conceived can be conceived to be greater than it is.
Thats absurd.
Perfection cannot exist in understanding alone, but must exist in reality.
Here is Descartes version:
"But, if the mere fact that I can produce from my thought the idea of something that entails everything that I clearly and distinctly perceive to belong to that thing really does belong to it, is not this a possible basis for another argument to prove the existence of God? Certainly, the idea of God, or a supremely perfect being, is one that I find within me just as surely as the idea of any shape or number. And my understanding that it belongs to his nature that he always exists is no less clear and distinct than is the case when I prove of any shape or number that some property belongs to its nature"
Godel's version (since I know you like Godel)
Definition 1: x is God-like if and only if x has as essential properties those and only those properties which are positive
Definition 2: A is an essence of x if and only if for every property B, x has B necessarily if and only if A entails B
Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified
Axiom 1: If a property is positive, then its negation is not positive.
Axiom 2: Any property entailed by—i.e., strictly implied by—a positive property is positive
Axiom 3: The property of being God-like is positive
Axiom 4: If a property is positive, then it is necessarily positive
Axiom 5: Necessary existence is positive
Axiom 6: For any property P, if P is positive, then being necessarily P is positive.
Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified.
Corollary 1: The property of being God-like is consistent.
Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.
Theorem 3: Necessarily, the property of being God-like is exemplified.
Plantinga's:
A being has maximal excellence in a given possible world W if and only if it is omnipotent, omniscient and wholly good in W; and
A being has maximal greatness if it has maximal excellence in every possible world.
It is possible that there is a being that has maximal greatness. (Premise)
Therefore, possibly, it is necessarily true that an omniscient, omnipotent, and perfectly good being exists.
Therefore, (by axiom S5) it is necessarily true that an omniscient, omnipotent and perfectly good being exists.
Therefore, an omniscient, omnipotent and perfectly good being exists.