Proof, Evidence and Arguments

[WinePusher]

1. Anyone who's taken high school geometry should be acquainted with the concept of a proof. An example of a mathematical proof (from trigonometry) is as follows:

A) secx (cotx) = cscx

B) 1/cosx (cosx/sinx) = cscx by the reciprocal and quotient identity

C) 1/[strike]cosx[/strike] ([strike]cosx[/strike]/sinx) = cscx by the rule that cross products cancel

D) 1/sinx = cscx by the reciprocal identity

A mathematical statement, theorem or postulate can be proven step by step with 100% precision. In otherwords, we are absolutely certain that secx (cotx) = cscx. It doesn't seem that this type of certainty exists anywhere else other than in mathematics.

2) Evidence and Arguments are complementary and tend to go hand in hand. The thing to note is that evidence that is nonmathematical and arguments in general cannot prove anything for certain. When using evidence and arguments the only thing we can establish is the likelihood and probability of an event.

Questions for debate:

1) When discussing issues like the existence of God, or the truth of evolution or the accuracy of the Bible, is it fair to say that none of these issues can be proven with absolute certainty? Will there always be room for doubt when it comes to God's existence and the theory of evolution?

2) Is there any mathematical proof for the existence of God? Does the existence of mathematics itself prove that God exists?

[Zzyzx]

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So, when discussing issues like the existence of God it is inappropriate to ask for proof.

I agree. The existence or non-existence of gods is devoid of proof.

However, claims of knowledge concerning gods CAN reasonably be challenged and the claimant be requested or required to provide substantiation for the claims. Weak to non-existence of substantiations (such as "I think so" or "He and that book say so" or "I feel something") are inadequate in debate.

[Cephus]

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So, when discussing issues like the existence of God it is inappropriate to ask for proof.

I agree. The existence or non-existence of gods is devoid of proof.

However, claims of knowledge concerning gods CAN reasonably be challenged and the claimant be requested or required to provide substantiation for the claims. Weak to non-existence of substantiations (such as "I think so" or "He and that book say so" or "I feel something") are inadequate in debate.

But it's not just claims of knowledge, claims of belief need to be challenged as well. Unfortunately, when it comes to religion, "faith" gets to be the "get out of reason free" card. So long as "faith" is allowed to be an excuse for not justifying your beliefs, what's the point in engaging in debate? One side can just make an unrealistic claim which cannot be backed up in any way.

Faith is pointless and useless in any debate, just declaring that you have faith that you're right doesn't actually mean anything.

[Zzyzx]

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Faith is pointless and useless in any debate, just declaring that you have faith that you're right doesn't actually mean anything.

I agree 100% and trust that readers differentiate between valid and invalid claims and "proofs."

When one resorts to a "faith argument" they have acknowledged a lack of reasoning and evidence.

Stating one's beliefs is proof of nothing at all other than the person's preferences.

[wiploc]

People keep saying that. I don't understand why. There are many non-mathematical things that we have complete confidence about. Do you have any doubt at all about whether "round" or "flat" is a better description of the earth's shape, about whether driving drunk at 90 miles an hour with no seat belt could be dangerous, about whether the sky sometimes seems blue to you, about whether homophobia is a thing, about whether some arguments purporting to prove the existence of god are poor, about whether there is a significant correlation between the latitude of one's ancestors and the color of one's skin, about whether tornadoes involve rotation, about whether ice floats, about which side of the road people usually drive on in your country, about whether the world ended yesterday, about where on your body you might find your fingernails?

The earth isn't round or flat, it's a sphere.

My point was that there are things (outside of mathematics) that we're sure of. You seem sure that the earth isn't flat, which endorses my position.

I'm sure there are plenty of professional drivers who could go 90 miles per hour with no seat belt and be fine

You say you're sure of that, which means that you agree with me.

If you still think you disagree with me, you need to read what I wrote again: "driving drunk at 90 miles an hour with no seat belt could be dangerous." Notice that your formulation made no mention of being drunk. Notice also that I used the word "could," as in "could be dangerous." That is to say, for me to be in error, danger would have to be impossible. There would have to be no circumstance in which driving drunk at 90 miles an hour with no seat belt would entail danger.

and there are lots of people who are color blind and who NEVER see the sky as blue.

Are you sure of that? Because if you're sure of that, then there are mathematical things that we are sure of, and you're agreeing with me.

Note also that where you reference blind people, I referenced only "you," my correspondent. If my correspondent happens to be colorblind, or totally blind, or locked away where he or she can never see the sky, then I could be wrong on this particular example. The other examples would still stand, and the claim that we are sure of some things non-mathematical will still be firmly defended.

We never have absolute certainty about any of those things.

Are you absolutely certain of that?

The only reason mathematics is absolutely certain is because we defined it that way.

Speaking of which, one of the definitions dictionary.com gives for "round" is "
spherical or globular, as a ball." So the reason we are absolutely certain that a round earth is the same as a spherical earth is that we defined it that way.

[Zzyzx]

.
Definitions are important in communication (and in thought).

Earth is spherical, meaning "Having a shape approximating that of a sphere."

"Sphere" is defined as: "a three-dimensional closed surface such that every point on the surface is equidistant from a given point, the center."

The Earth is actually an oblate spheroid (or ellipsoid of revolution) – with a polar diameter shorter than its equatorial diameter (by about 27 miles).

[WinePusher]

I'd give you MPG points, but I have none left to give . Excellent post. Far too often I see people -- on both the theist and non-theist sides of the debate -- touting "proof" that god(s) exist/don't exist. This is ridiculous, as there are very few propositions (mathematical statements, tautologies, etc.) that can be known for certain. For everything else (including religious and scientific claims), we must infer to the best explanation using a combination of empirical evidence and rational argument.

Anyone who says "I am certain God exists/doesn't exist" is speaking foolishness.

Yup thanks for making my point better than I could!

2) Is there any mathematical proof for the existence of God? Does the existence of mathematics itself prove that God exists?

This question requires a more robust response. I'll address each sub-part separately.

• A. Some theists claim that the scholar Kurt Gödel's ontological proof constitutes mathematical proof for God's (I'll use this spelling of "God" [and the neutral pronoun "it"] from now on, as we are talking about the specific deity concept from classic philosophical theism) existence. Gödel's proof, which uses modal logic, is:
  

  Definitions:
  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
  
  Axioms:
  Axiom 1: Any property entailed by—i.e., strictly implied by—a positive property is positive
  Axiom 2: A property is positive if and only if its negation is not 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 a positive property
  
  Theorems:
  Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified.
  Theorem 2: The property of being God-like is consistent.
  Theorem 3: If something is God-like, then the property of being God-like is an essence of that thing.
  Theorem 4: Necessarily, the property of being God-like is exemplified.[/color]

  Gödel's proof--like all versions of the ontological argument--fails because it is based upon flawed definitions and axioms. Specifically, his notion of "positive properties" is incoherent because it relies upon a set of purely subjective value judgments. That is, who is to say that, for example, the "positive property" of knowledge is objectively better than the "non-positive property" of ignorance? It is only due to our human subjectivity that we value knowledge over ignorance.
  
  In addition, Axiom 5 is absurd because existence (necessary or otherwise) is not itself a property, but is a precondition for possessing any properties. For some "A" to have any properties, "A" must first exist.
  
  As this (and other very similar proofs like the philosopher Alvin Plantinga's modal ontological argument) is the only attempt at a mathematical case for God's existence, I think it's safe to say that there exists no successful mathematical proof for God's existence.
  
  B. No, the existence of mathematics does not act as evidence for God's existence. As others have mentioned, the human-created mathematical concepts of numbers, ratios, and so on are based on objective properties of the universe. In other words, they are but descriptions of the ways in which the universe behaves. They aren't necessarily true or prescriptive in any sense, but purely descriptive.
  
  One could, for example, devise a precise set of descriptive statements for the functioning of an evolved software program, but that does not imply that the program was created or designed by any mind.

I have never studied anything by Kurt Gödel so I can't really comment on this. I've seen you debate this subject with EduChris many times in the past so I'll leave this to you and him.

My argument is that the existence of mathematics itself provides a good indication that God exists. The reasons for this are as follows:

Mathematics is the language of the universe. Mathematics is itself rational and consistent, and the fact that the universe can be expressed in this rational and consistent language should make us wonder what exactly is holding up mathematics to begin with. In other words, the universe is based upon mathematics, so what exactly is mathematics based upon? I'm not sure I've heard any naturalistic explanation to this question. Perhaps you can provide one? What makes mathematics true and what accounts for the origins of mathematics?

These questions pertaining to mathematics seem to be just as baffling as many of the question pertaining to consciousness. On a naturalistic worldview the existence of both mathematics and consciousness seem un-explainable.

[Inigo Montoya]

WinePusher wrote


My argument is that the existence of mathematics itself provides a good indication that God exists. The reasons for this are as follows:

Mathematics is the language of the universe. Mathematics is itself rational and consistent, and the fact that the universe can be expressed in this rational and consistent language should make us wonder what exactly is holding up mathematics to begin with. In other words, the universe is based upon mathematics, so what exactly is mathematics based upon? I'm not sure I've heard any naturalistic explanation to this question. Perhaps you can provide one? What makes mathematics true and what accounts for the origins of mathematics?

These questions pertaining to mathematics seem to be just as baffling as many of the question pertaining to consciousness. On a naturalistic worldview the existence of both mathematics and consciousness seem un-explainable

The existence of mathematics? Is mathematics an object or a concept? Is mathematics physically and presently occupying a point in space? Does it have form? What is this ''God?'' Does it have form? Does it presently and physically occupy a point in space? How is it related to our species' invention and use of numbers?

Words have to mean something unambiguous in dialogue of this scale. To say that mathematics is the language of the universe is to say that the universe has a language.... and it is mathematics. In what way is the ''the universe based on mathematics?'' Are you sure that isn't backwards? Are you sure we aren't instead using the mathematics we invented in an attempt to explain the universe?

Please define the following:

Exist _________
God __________
Absolute Certainty _________

Can you ''prove'' you ''exist'' with ''absolute certainty?''

[1213]

My reason for saying this is because, IMHO, the Biblical fables of God contradict the very character and essence of what the God is supposed to represent in the first place. (i.e. the Highest form of intelligence, omniscience, and omnipotence)

That is interesting idea. Are you really able to determine highest intelligence? What do you think, are you really determining your intelligence the most high?

[Cephus]

People keep saying that. I don't understand why. There are many non-mathematical things that we have complete confidence about. Do you have any doubt at all about whether "round" or "flat" is a better description of the earth's shape, about whether driving drunk at 90 miles an hour with no seat belt could be dangerous, about whether the sky sometimes seems blue to you, about whether homophobia is a thing, about whether some arguments purporting to prove the existence of god are poor, about whether there is a significant correlation between the latitude of one's ancestors and the color of one's skin, about whether tornadoes involve rotation, about whether ice floats, about which side of the road people usually drive on in your country, about whether the world ended yesterday, about where on your body you might find your fingernails?

The earth isn't round or flat, it's a sphere.

My point was that there are things (outside of mathematics) that we're sure of. You seem sure that the earth isn't flat, which endorses my position.

But we have evidence to support the shape of the planet. We cannot be absolutely certain but we have a lot more evidence to support it today than we did 2000 years ago.

I'm sure there are plenty of professional drivers who could go 90 miles per hour with no seat belt and be fine

You say you're sure of that, which means that you agree with me.

All it takes is a single example to prove my point.

If you still think you disagree with me, you need to read what I wrote again: "driving drunk at 90 miles an hour with no seat belt could be dangerous." Notice that your formulation made no mention of being drunk. Notice also that I used the word "could," as in "could be dangerous." That is to say, for me to be in error, danger would have to be impossible. There would have to be no circumstance in which driving drunk at 90 miles an hour with no seat belt would entail danger.

Could be dangerous? To who? You don't define your terms. Sure, for most people it could be dangerous. That doesn't mean it is dangerous, just could be. You're hedging.

and there are lots of people who are color blind and who NEVER see the sky as blue.

Are you sure of that? Because if you're sure of that, then there are mathematical things that we are sure of, and you're agreeing with me.

Note also that where you reference blind people, I referenced only "you," my correspondent. If my correspondent happens to be colorblind, or totally blind, or locked away where he or she can never see the sky, then I could be wrong on this particular example. The other examples would still stand, and the claim that we are sure of some things non-mathematical will still be firmly defended.

We never have absolute certainty about any of those things.

Are you absolutely certain of that?

There's no such thing as absolute certainty about anything, including mathematics.

[wiploc]

People keep saying that. I don't understand why. There are many non-mathematical things that we have complete confidence about. Do you have any doubt at all about whether "round" or "flat" is a better description of the earth's shape, about whether driving drunk at 90 miles an hour with no seat belt could be dangerous, about whether the sky sometimes seems blue to you, about whether homophobia is a thing, about whether some arguments purporting to prove the existence of god are poor, about whether there is a significant correlation between the latitude of one's ancestors and the color of one's skin, about whether tornadoes involve rotation, about whether ice floats, about which side of the road people usually drive on in your country, about whether the world ended yesterday, about where on your body you might find your fingernails?

The earth isn't round or flat, it's a sphere.

My point was that there are things (outside of mathematics) that we're sure of. You seem sure that the earth isn't flat, which endorses my position.

But we have evidence to support the shape of the planet. We cannot be absolutely certain but we have a lot more evidence to support it today than we did 2000 years ago.

You are in doubt about whether the world is flat?

All it takes is a single example to prove my point.

But you're not sure of that?

If you still think you disagree with me, you need to read what I wrote again: "driving drunk at 90 miles an hour with no seat belt could be dangerous." Notice that your formulation made no mention of being drunk. Notice also that I used the word "could," as in "could be dangerous." That is to say, for me to be in error, danger would have to be impossible. There would have to be no circumstance in which driving drunk at 90 miles an hour with no seat belt would entail danger.

Could be dangerous? To who? You don't define your terms. Sure, for most people it could be dangerous. That doesn't mean it is dangerous, just could be. You're hedging.

Of course I'm hedging. I'm making the point that if we can't know a particular thing (that the earth isn't flat, say) then we can still know that it is probable (I guaran-darn-ty you that the earth probably isn't flat.) If we know that the earth probably isn't flat, then it is possible to know things.

We never have absolute certainty about any of those things.

Are you absolutely certain of that?

There's no such thing as absolute certainty about anything, including mathematics.

Which proves the point I already made: The only logical way to argue that we don't know anything except mathematics is to adopt a skepticism so radical that we don't know even mathematics.