Axioms and Non-Euclidean Geometry

Posted to LDL on 20. November, 1997

Amanda Maxham asks a number of questions about axioms and the implications of non-Euclidean geometry on their validity:

Does the axiom "existence exists" have anything to do with the real world,or is it just an arbitrary starting point? If it does, how do we know that it corresponds?

Quite literally, it has everything to do with the real world (do you see what I mean?). To call it "arbitrary" is to commit just about the worst stolen concept fallacy of which I can think: A major precondition for criticizing the arbitrary is the acceptance of the existence of the real world to which judgments must correspond.

Can axioms be internally consistent, yet not correspond to reality?

No, they need to be perceptually self-evident, too. Existence exists is perceptually self-evident. Just look at all of the things around you.

Are axioms arbitrary because they can not be proven? Why? I mean are they an arbitrary starting point that could have been otherwise?

What I said about arbitrariness and the stolen concept applies to this as well. Actually, your connection between arbitrariness and the possible contingent nature of existence as such brings out an interesting point here: The idea that "existence exists" could have been otherwise is presupposing a primacy of consciousness viewpoint. To say that something could have been otherwise is to speak of it as a product of someone's choice. Whose choice are we talking about here? God's? But POC just doesn't work. There has to be existence first for anyone to be conscious of it. Existence could not have been otherwise -- it just is.

The example of Euclidean geometry was brought up. Euclid started on his own "axioms", or rules, which can't be proven (they are starting points). Anyway, I am not too sure about this because I really don't know that much about Euclid etc. The point was that if we started with other axioms, (non-euclidian geometry), we would come to different conclusions. But how can we tell which ones correspond to reality if the starting points are arbitrary? I have heard that Euclidean geometry is not correct according to what we've discovered in the 20th century.

Oh I have so many things to say. First of all, listen to Harry Binswanger's "Selected Topics in the Philosophy of Science". I believe that your club has it in its library. Harry gives an excellent argument against this argument from non-Euclidean geometry. I'll summarize a number of things I can remember from it.

First of all, Euclidean axioms properly defined are not arbitrary starting points. They deal with actual, observable properties of our immediate physical context. Dr. Binswanger notes that several of the definitions traditionally associated with Euclid are probably not the best, so he offers better ones.

Next, Euclidean axioms of geometry are very different sorts of creatures than metaphysical axioms. They're just not about the same things, for instance. Whereas metaphysical axioms are the starting points for all knowledge, geometrical ones are starting points for a very specific and narrow discipline. Although they do not presuppose any other geometrical concepts, they do presuppose many other philosophical ones. So while they involve perceptual self-evidencies, they are not irreducible in the same sense in which metaphysical axioms are. So even if they are, in some sense, wrong, there is not sufficient analogy between them and metaphysical axioms to suggest that their being wrong somehow negates the metaphysical axioms.

Also, note carefully one of your sentences: "The point was that if we started with other axioms,(non-euclidian geometry), we would come to different conclusions." This argument is true. However, when it is used to somehow deny the validity of metaphysical axioms, it is committing yet another stolen concept fallacy. What is the concept it steals? The concept of "logical inference". The argument notes that if you select certain assumptions, different conclusions follow logically from them. But to use this to deny the metaphysical axioms, which include the Identity axiom which is the basis of logic, is a conceptual grand larceny.

As for Euclid's being wrong, this is an outright lie. The non-Euclidean geometries of Riemann, et al, are different insofar as they alter Euclid's second(?) postulate about parallel lines. This is because it appears that parallel lines take on different properties on curved surfaces than they do on flat ones. But Euclid's geometry was only meant to apply to flat surfaces in the first place. That was the context he was assuming. Non-Euclidean geometry has merely expanded the application of geometry to new contexts -- it hasn't destroyed its own basis, Euclidean geometry.

If every is implies and ought, does that make all actions moral actions? If not, what criterion distinguishes moral actions from non-moral actions?

All chosen human actions are susceptible to moral judgment, if that's what you mean. I don't quite understand your second question.

I don't understand if it is justified to say that some views aren't philosophical.

Here's a non-philosophical view: There is a computer in front of me. Now, philosophy might have certain things to say about such views, such as that they are valid as against what a skeptic might say, but philosophy can never dictate the particular content of such views. The same goes for the particular content of any science. Philosophy studies the fundamental nature of the universe, man and his relationship to the universe. Some topics are important (to some people), but they're not fundamental as far as knowledge qua knowledge is concerned.

Oedipus is morally responsible for killing another man, but is he morally responsible for killing his father, even though he didn't know that the man he killed was his father? Why? What does it mean to be morally responsible for something?

Well I don't see whether it was his father or not matters to the moral judgment. To be morally responsible for a consequence is merely to have made a choice which resulted in that consequence, when that consequence has some value-significance for someone.

He is morally responsible for killing his father, because his belief in the matter doesn't alter the fact that it was his father (although he might not know this). The issue of proper names and how they refer in such contexts is an issue which philosophers of language love to complicate, but it's really not very interesting.

Can the ultimate value (life) be called good in itself? Is the ultimate good still good to something, for something?

I asked myself this question once, and I think that the answer is actually pretty easy. The reason there is some confusion at first is because Objectivists call life an end-in-itself, but other philosophers typically treat "end-in-itself" and "intrinsic good" as interchangeable. But they're not. Yes, even an end-in-itself is still a value to someone, for something. Life is a value....to the organism which possesses it. This means: It's an end-in-itself, but it's not an intrinsic value.

Does life as the ultimate value imply an infinite regress because it is self-sustaining and self-generated action?

I don't understand the question.

What was it that you said during the logic lectures about Aristotle's doctrine of the means? That he assumes the standard (the middle) before he deems the "extremes" to be extreme? Can you explain that again?

Well, his method essentially consists of pointing out what the two extremes are and noting that they are both bad. Then he says that we can know what the good is because it is in between. My question was: How did he know that they extremes were bad if he hadn't already defined the good? He couldn't know as much. This means that he had already subconsciously formulated what he wanted to be in the middle. So his argument is really more of a clever illustration than it is a demonstration. He just happened to pick some nice, egoistic virtues in doing so.

Is the "causal likeness principle" true? As in in order for something to cause an action in another thing, do they have to be of the same nature?

I don't understand why anyone would think this. I am a very different kind of thing than that which I am using to type these words into a computer. I am a living organism, and it is compose of insentient matter. And yet I seem to be typing...