Something To Prove

[DBCA]

Last week in math class (I'm a freshman at St. John's College) we were demonstrating and discussing some propositions from Euclid's Elements when I made the comment that the propositions were not proofs per se, because they were derivations of conclusions from a set of definitions, postulates, and common notions, and not self-evident axioms. The tutor (we call our professors "tutors") asked me what a "self-evident axiom" would be, I told him that "existence exists"; he seemed skeptical . Anyway, wonder of wonders today he gave me an assignment to "prove something" it doesn't have to be mathematical, just anything. I want to keep it simple, my first idea was to just prove that the axioms of Objectivism are in fact axioms (i.e.- that you have to accept them before you can deny them). I'd like your suggestions, and, since I must have the proof thought out and typed up (to hand out after copying) by Wednesday 1:00 PM EST, your help. It is a very informal thing, I won't be graded on it so don't worry that you cheating for me (for a fuller explaination of why help on this wouldn't be cheating, see "Who is the Final Authority in Ethics" in VOS I believe). Thanks y'all!

[andrew]

I'd be careful when it comes to proving an axiom. The point with axioms being self-evident is that they can't be proven since proof relies on those axioms; they presuppose it. Axioms can be validated[/], voicing your point that in order to deny them one has to accept them.

I think an interesting proof to check out is the one for Ayn Rand's theorem "Egoism as rational moral value of identity"

[DBCA]

yeah, I know that the axioms can't be proven to be true...What I was going to do would be to prove that they were axioms, thanks though.

[danielshrugged]

You could prove that the axioms are in fact axioms (in other words, validate them), but I would advise against this. It's the most abstract and one of the most controversial proofs you could select. I would recommend something simple and non-philosophical. The first possibility that occurs to me is the law of inertia (this would require experiment, in part).

[DBCA]

Daniel,

I think what you say is true, that it is abstract and controvertial, but that is really what I want in this context. Firstly, one of the drugged-up communists in my class has everyone pretty well convinced that geometry has nothing to do with reality because it is so abstract. He's convinced everyone that it is just arbitrary, I want to demonstrate to them that abstract does not mean arbitrary and that there are rules (such as non-contradiction) that must be observed. Secondly, if it wasn't controvertial I'd have little interest in it; I already know that things can be proven, I just want to sock it to the skeptics. What do you think?

[tommyedison]

You could prove reason as man's only guide and the dangers of relying on faith.

You could qoute Thomas Jefferson:-

"Man once surrendering his reason, has no remaining guard against absurdities the most monstrous, and like a ship without rudder, is the sport of every wind."

By the way is "existence exists" really an axiom? Let's say that our existence is an illusion. But there must be something (us) that perceives the illusion. Thus we do exist. Any other belief leads to a self-contradiction.

[danielshrugged]

You could prove reason as man's only guide and the dangers of relying on faith.

You could qoute Thomas Jefferson:-

"Man once surrendering his reason, has no remaining guard against absurdities the most monstrous, and like a ship without rudder, is the sport of every wind."

By the way is "existence exists" really an axiom? Let's say that our existence is an illusion. But there must be something (us) that perceives the illusion. Thus we do exist. Any other belief leads to a self-contradiction.

Damn. I thought I was through reading Descartes for the year.

[Godless Capitalist]

one of the drugged-up communists in my class has everyone pretty well convinced that geometry has nothing to do with reality because it is so abstract. He's convinced everyone that it is just arbitrary, I want to demonstrate to them that abstract does not mean arbitrary and that there are rules (such as non-contradiction) that must be observed.

Does he really mean arbitrary in the sense of not having to follow any logical rules? Or does he just mean that geometry is a mathematical model that may or may not correspond to something in reality? (eg Euclidean geometry can be quite useful in building a house)

[stephen_speicher]

Firstly, one of the drugged-up communists in my class has everyone pretty well convinced that geometry has nothing to do with reality because it is so abstract....  I just want to sock it to the skeptics.

In that case you might want to consider a little-known yet popular concept among some mathematicians known as "Proof Without Words." These consist of purely geometric constructions which prove various mathematical notions without explicit language. There is nothing quite like a really good strictly geometric proof of some fact of reality to shake loose the notion that geometry itself is not connected to reality. There are dozens of these sorts of geometric proof constructs, ranging from proving the distributive property of a triple scalar product, to every cube being the sum of consecutive odd numbers, to integration by parts.

If you are interested in doing something like this, let me know.

[DBCA]

sorry for not responding sooner, ive written up a first draft of the axiom proof. I'm going to present it on friday so I'd really appreciate any comments you all might have. I think the identity section is a too sparse and perhaps the fallacies section could be more thorough.

The Axioms of Metaphysics

By Donald B.C. Allen

“An axiomatic concept is the identification of a primary fact of reality, which cannot be analyzed, i.e., reduced to other facts or broken into component parts. It is implicit in all facts and in all knowledge. It is the fundamentally given and directly perceived or experienced, which requires no proof or explanation, but on which all proofs and explanations rest.”1

“What is self-evident need not be evident at once, or to everybody; the intelligible is intelligible only to the intelligent. In calling anything self-evident we mean not that it is evident without need for understanding, but that we need consider nothing but the terms of the judgment, to see its necessity.”2

The following is a proof that the ideas “existence”, “consciousness”, and “identity” are axioms, that is to say, that they are irreducible primaries at the base of all knowledge. The proof consists of making explicit the fact that it is impossible to reject these ideas, because in any attempt to deny them one must first implicitly accept them. This is not a proof that the axioms are true per se, only that them are axioms.

Axiom I: Existence

“The concept of ‘existence’ is the widest of all concepts. It subsumes everything—every entity, action, attribute, relationship (including every state of consciousness)—everything which is, was, or will be.”3

Existence exists; something, as against nothing, is. By denying that there is something and asserting, “no, there is nothing” one implies a subject matter about which something is trying to be determined. The objection requires that there be something in question about which one can disagree, an untrue statement about that thing to object to, as well as the objection's own existence.

Note: Axiom I says nothing about the specific nature of existence, e.g.- that it is physical or otherwise, it says not what it is but only that it is.

Axiom II: Consciousness

The act of grasping Axiom I implies a second axiom, that you exist possessing consciousness, consciousness being the faculty of awareness of existence. It is impossible to argue against this axiom because doing so would necessarily involve making purportedly true statements about existence. Even making a sophistic refutation of this axiom would imply that the objector is conscious of existence, the fact that it has a specific nature, and the notion that the rejected statement does not represent this nature i.e.—that it is untrue.

“A consciousness conscious of nothing but itself is a contradiction in terms: before it could identify itself as consciousness, it had to be conscious of something.”4

Axiom III: Identity

To exist is to exist as something. To reject that existence has a specific nature, is to identify existence as something, that something being a “nothing-inparticular” which nonetheless is an identification, albeit a fallacious one.

Notes on Some Fallacies of Knowledge

If one argues that ideas are in some way true and false at the same time and in the same respect, one assumes that this property does not apply to itself. That is to say, the idea that “ideas can be true and false at the same time and in the same respect” is absolutely and universally true, in effect negating itself. Likewise, if one asserts that that “nothing can be known about existence” then one upholds this idea as knowledge about existence. To assert that “nothing can be known about existence, but this idea itself is only true for me and may not be so for anyone else” implies that it is possible to know the pluralistic nature of knowledge.

Endnotes:

1. Introduction to Objectivist Epistemology, Ayn Rand

2. An Introduction to Logic, H.W.B. Joseph

3. Objectivism: The Philosophy of Ayn Rand, Leonard Peikoff

4. Atlas Shrugged, Ayn Rand

[BurgessLau]

Here is a question to consider as preparation not only for the class but also for your own long-term knowledge:

What does "prove" mean?

In other words, in general terms, what constitutes a proof of anything?

P. S. Convincing someone means leading him to knowledge. Is it possible to convince philosophical skeptics of anything? By definition, they know nothing and therefore learn nothing -- not even that they can know nothing. Sextus Empiricus (lived around 200 CE) said that he didn't know whether he knew anything, but he only had feelings and followed the customs of his neighbors. (See Julia Annas, translator, Sextus Empiricus: Outlines of Scepticism.) Thus, skepticism is merely emotionalism and traditionalism.

[danielshrugged]

I strongly advise against quoting Ayn Rand, Leonard Peikoff, or anyone else, for that matter, in this paper. If you have to quote others to support your point, it means you don't fully understand the proof. Note that a proof does not require knowledge of what others have said about the matter. Also, Ayn Rand has no place in a freshman mathematics paper (however inappropriate this assignment may be). You're studying Euclid, not Objectivism.

Even allowing for your quotes, I do not think you've proved that the axioms are axioms. You might choose to focus just on one of the axioms so that you have time to develop one proof, rather than the three you are attempting now.

As stated, your proof of existence is no more convincing than this: God exists. By denying His existence, one denies the power which makes all action possible, which denial of action is contradicted in the action of denial.

It is not at all clear why the denial of each axiom implies what you say it implies. Nor is it clear how your concepts of the axiomatic and the self-evident originate in perception. As stated, they sound like arbitrary starting points.

[dougclayton]

You could prove reason as man's only guide and the dangers of relying on faith.

You could qoute Thomas Jefferson:-

"Man once surrendering his reason, has no remaining guard against absurdities the most monstrous, and like a ship without rudder, is the sport of every wind."

I wouldn't say this proves anything about reason. It's very helpful to concretize reason's absence, but an analogy is not a proof.

By the way is "existence exists" really an axiom? Let's say that our existence is an illusion. But there must be something (us) that perceives the illusion. Thus we do exist. Any other belief leads to a self-contradiction.

Ignoring the Cartesian implications that Daniel rightfully pointed out, what you just described is what makes it an axiom. Anyone who says something is an illusion (including all of existence) is tacitly relying on the fact that there are things that exist that can be perceived, and therefore misperceived (an "illusion"). But that is what they are trying to deny. And the reason that "existence exists" is an axiom is because every argument against it has the same circularity. (That fact, by the way, is how you prove it is an axiom.)

I wouldn't phrase it as "any other belief leads to a self-contradiction," because the lack of contradiction (with reality, not with your own beliefs) is what makes a statement right, not what makes it an axiom.

[dougclayton]

I agree with others that you've bitten off quite a bit trying to prove the axiomatic nature of existence (to say nothing of consciousness and identity). One thing bothers me, though: the striking similarities between your proof and the phrasing of both Dr. Peikoff and Ayn Rand. The fact that you use the same sequence of words at several different points suggests that you have not grasped the argument as much as you have memorized it. Some examples (all emphasis mine):

The following is a proof that the ideas  existence, consciousness, and identity are axioms, that is to say, that they are irreducible primaries at the base of all knowledge. The proof consists of making explicit the fact that it is impossible to reject these ideas, because in any attempt to deny them one must first implicitly accept them.  This is not a proof that the axioms are true per se, only that them are axioms.

The foregoing is not a proof that the axioms of existence, consciousness, and identity are true.  It is a proof that they are axioms, that they are at the base of knowledge and thus inescapable.

--------------------------------------

Existence exists; something, as against nothing, is.

All of these [men] know equally the fundamental fact that there is something, something as against nothing.

--------------------------------------

Note: Axiom I says nothing about the specific nature of existence, e.g.- that it is physical or otherwise, it says not what it is but only that it is.

This axiom does not tell us anything about the nature of existents; it merely underscores the fact that they exist.

--------------------------------------

The act of grasping Axiom I implies a second axiom, that you exist possessing consciousness, consciousness being the faculty of awareness of existence.

Existence exists--and the act of grasping that statement implies two corollary axioms: that something exists which one perceives and that one exists possessing consciousness, consciousness being the faculty of perceiving that which exists.

This reminds me of very good advice I read many years ago on OSG: you'll know you understand it when you find your own voice. A good example here, ironically, is OPAR: in re-reading it looking for the matching phrases, I came across all the concretization he used (tomatoes, for instance, instead of the abstract term "existents"). In contrast, the passages in your proof which don't seem to sound like OPAR are very "abstract" (in the sense of "floating")nvoluted:

teBegin-DBCA+Sep 28 2004, 03:36 AM-->

By denying that thereethinsserting, "no, there is nothing" oieject matter about which something is trying to be determined. The objection requires that there be something in question about which one can disagree, an untrue statement about that thing to object to, as well as the objection&/b]stenteEnd-->

This will not convince anyone but those who already agree with you or who are very interested in metaphysics. Everyone else will see this as just as arbitrary as any other philosophical argument.

[danielshrugged]

An additional point: What would your tutor think if, in respnse to his or her request that you prove something, you turned in a paraphrase of one of Euclid's proofs? Well, what have you done here but turn in a paraphrase of Ayn Rand?

[DBCA]

Doug, these are good points, now that I've looked all this over I'm a little embarrassed. I can see that I was being overzealous and I really need to study the subject of axioms more before I undertake something like this.

stephen_speicher mentioned a "proof without words", I'm very interested in that and like very much to hear more about it.

Also, Daniel, I can't see what else there would be to the existence axiom; can you recommend some more in-depth reading on that?

Thanks everybody

[MisterSwig]

Anyway, wonder of wonders today he gave me an assignment to "prove something" it doesn't have to be mathematical, just anything.

Really, it is not that difficult to prove something. If your "tutor" wants a deductive proof, then give him one he can't possibly deny without revealing himself as a true blue skeptic.

All killer whales have a blowhole.

Shamu is a killer whale.

Therefore, Shamu has a blowhole.

If your "tutor" wants an inductive proof, then show him a bunch of pictures of whales and say, "Look, they all have blowholes."

If your "tutor" denies the simplest induction, then write "Paper Exists" on a piece of paper and hand it to him.

If he denies the existence of the paper in his hand, then give up. Start discussing the validity of the senses with him.

All this "tutor" wants you to do is prove "something." If you can't get him to accept the simplest proofs, then what is the point of performing some elaborate deduction or induction for him?

Anyway, that is my take on this situation.

[danielshrugged]

Really, it is not that difficult to prove something. If your "tutor" wants a deductive proof, then give him one he can't possibly deny without revealing himself as a true blue skeptic.

All killer whales have a blowhole.

Shamu is a killer whale.

Therefore, Shamu has a blowhole.

If your "tutor" wants an inductive proof, then show him a bunch of pictures of whales and say, "Look, they all have blowholes."

If your "tutor" denies the simplest induction, then write "Paper Exists" on a piece of paper and hand it to him.

If he denies the existence of the paper in his hand, then give up. Start discussing the validity of the senses with him.

All this "tutor" wants you to do is prove "something." If you can't get him to accept the simplest proofs, then what is the point of performing some elaborate deduction or induction for him?

Anyway, that is my take on this situation.

I agree with the essence of this. If the assignment is about proof as such, simpler is better. (I disagee that you can prove that all whales have blowholes simply by looking at pictures of whales.)

[danielshrugged]

Also, Daniel, I can't see what else there would be to the existence axiom; can you recommend some more in-depth reading on that?

I don't know of any reading on the subject. But you can start by asking yourself where, aside from Ayn Rand, you got your concepts of axiom and of self-evident. You should be able to point ultimately to objects of perception to answer this question.

[stephen_speicher]

stephen_speicher mentioned a "proof without words", I'm very interested in that and like very much to hear more about it.

Just to be clear, I suggested it as a possibility because you wanted to "sock it to the skeptics" in your class who claimed "that geometry has nothing to do with reality because it is so abstract." This "proof without words" uses purely geometric constructions to prove various mathematical propositions and formulas, everything from proving the distributive property of a triple scalar product, to every cube being the sum of consecutive odd numbers, to integration by parts.

If you are interested I can send you several examples of this. I doubt that you have the mathematical sophistication to come up with one of your own, so you would have to present any one that you choose by reference to the source from whence it came. You would also have to be sure that you understand the geometrical construction and just how the geometry proves the proposition. So, even though this would not be an original proof of your own, you would still have to struggle to grasp how just how it is done. And, since this is a math class assigment this would be on topic.

If you do want to pursue this approach then write to me privately at my address below and note your email address so I can send you several pdf files with different geometric constructions. (Note that these are favorite pastimes of a small group of mathematicians and these have been published fairly regularly for a while in a mathematical journal.)

[nimble]

i know why your professor did this. He wants to show you that things are really hard to prove. Mathematics is about the only subject where things can be "proven." All observational science is inductive and therefore not proof. And math relies on definitions, the exact notion that you critiqued him about. So i would be very careful, in fact, i think it would be best to just not challenge him here...think economically does the benefits outweigh the costs?

[danielshrugged]

i know why your professor did this. He wants to show you that things are really hard to prove. Mathematics is about the only subject where things can be "proven." All observational science is inductive and therefore not proof. And math relies on definitions, the exact notion that you critiqued him about. So i would be very careful, in fact, i think it would be best to just not challenge him here...think economically does the benefits outweigh the costs?

I'm not so sure that's the tutor's motive...which tutor is this, anyway?

[nimble]

I'm not so sure that's the tutor's motive...which tutor is this, anyway?

What do you think the motive was? I think the professor didnt like that a student critiqued him, so he was going to let the student try to prove something, then critque him.

[danielshrugged]

What do you think the motive was? I think the professor didnt like that a student critiqued him, so he was going to let the student try to prove something, then critque him.

Ummm, but this student didn't critique the tutor (it was the other students who apparently disagreed with him); it's not clear whether the student and tutor disagree. Also, St. John's is all about encouraging independent thought. No tutor gets upset just because someone disagrees.

[PirateF]

If I may ask, assuming you had an understanding of Objectivism when you were selecting a school...what made you choose St. John's?

The reason I'm asking is, my sister graduated from there. Not that she had much of a grasp on reality before she went, but in her four years there I watched her struggle so much because of the lack of clear direction, the "we don't claim to know any more than you" mentality. I sat in on a lecture..whoops, excuse me, a seminar - on John Locke, and was horrified that they spent three hours discussing his words in a complete vaccuum. Each time someone would attempt to include some fact that would lend context to his writing, they would get a slap on the wrist and a reminder that they were only allowed to discuss the words in this specific essay (sorry, can't recall which one). After the class I mentioned my concern about the lack of context and my sister's response was a nonchalant, "Yeah, we tend to avoid context here."

Anyway, I definitely see the appeal to studying the "Great Books" in depth, but the discussion method used at St. John's definitely troubles me.