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Hilbert's Axioms

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Posted by Harvey on December 28, 2001 21:00:43 UTC


Let me use Hilbert's axioms since he is well known for his attempt to restate Euclid's attempts by intentionally using undefined terms. There's a number of axioms, so I'll just state the axioms of incidence:

Axioms of Incidence

1) For every two distinct points, there is at least one line on both of them.

2) For every two distinct points, there is at most one line on both of them.

3) Every line has at least two points on it. There exist at least three points not on the same line.

4) Through any three points not on the same line, there is one and only one plane.

The 'official' undefined terms used are: point, line, plane, on.

However, every word is undefined if you look at the above terms strictly from a mathematical (formal) perspective. If we do as you suggest and state that any undefined term is utterly devoid of meaning, then the entire set of axioms are devoid in meaning. Mathematics becomes devoid of any meaning.

Now, if we look at the 'official' undefined terms (point, line, plane, on), what distinguishes these terms is that they are reference terms. That is, in normal communication about the world we should be able find their reference with some physical object. This is what separates mathematics from discussion about physical objects. We are talking about mathematical concepts. Normally mathematical concepts require mathematical descriptions, but since these terms are primitive they cannot be described in terms of other mathematical descriptions. So, how do these terms acquire meaning?

Firstly, the terms already have some meaning because they are phrased in a language we understand. Take the word 'on'. On already has meaning in the English language. So we already know what the axiom (1) means when it states "For every two distinct points, there is at least one line on both of them." In fact, we also know what the whole axiom means since it is phrased in the English language which is a language that we understand and therefore we can understand the meaning of the axiom. If we couldn't understand English, then we couldn't understand what this axiom was talking about.

Secondly, the language context mentioned in the above listed axioms help us understand the English terms better by clarifying what is meant by point, line, plane, on, etc. For example, axiom (1) helps us to visualize how at least one line can be placed on two points (i.e., we already have an idea of what a point and line is in the English language and the wording of the axiom allows us to visualize what the axiom is stating). Similarly, axiom (4) fills in our understanding of the mathematical meaning of a plane. We already have some concept of what a plane is, but seeing the term 'plane' used in context adds further to the meaning so that if we have any extraneous notions of what a plane is we can see that these notions should be discarded.

>>> the context of mathematics, the terms are considered to be totally devoid of meaning. As a consequence, the statements that use them are also totally devoid of meaning. They lack total meaning.>In the context of mathematics, the term 'set' has no meaning, whether you want to call it formal meaning or not. In the context of mathematics, it is not obvious at all that a set means a collection of members. The term 'collection' does not appear, so it is not even defined in that context. The term 'member' is also taken as undefined, as is the concept of 'belonging'. So, in the context of mathematics we can talk about an element belonging to a set, but what that means is completely undefined, unknown, and unimportant.>>You are confusing the definition of a particular set with the definition of the term 'set'. What you have described are methods of defining particular sets. This is done, and must be done, all the time in mathematics otherwise there would be nothing to talk about. As an example, I can define a set A as containing the members p and q. That is a perfectly good, mathematically acceptable, definition of the set A. And, I suppose you could say that the set A now "means" the pair p, q. I am not sure that is what you mean by 'mean', but it is the closest you can get to meaning in mathematics. But really, saying "the set A contains p and q" means absolutely nothing. It sets some constraints, however. For example we can say with mathematical certainty that the cardinality of A is 2. But we can say that only because we have agreed to say it. There is still no meaning.>In the preceding paragraph, I talked about what might be the meaning of a particular set, A. And you are right, after making such a definition, constraints emerge which suggest certain meanings which we are tempted to ascribe to, for example, the set A. But even this is not what we were talking about earlier. We were talking about the undefined terms themselves, not particular instances of these terms. I.e. we were talking about the term 'set', not about a particular set, A. While in the process of some mathematical development, a set, such as A, may seem to begin taking on meaning, the undefined term 'set' does not.>Or, if it seems to, mathematical rigor demands that such notions be kept strictly out of the formal development>H: In the philosophy of mathematics there is a great deal of discussion on the actual meaning of sets and set theory, but this is philosophy since mathematicians are only concerned with working within set theory without consideration of the philosophical meaning of sets as they apply to the world. P: This exemplifies exactly the point I am trying to make. A discussion of meaning is appropriate in the context of philosophy, but it has absolutely no place in the context of formal mathematics.>>H: Terms must mean something otherwise mathematicians would be unable to distinguish a group from a set, or a set from a point, or a point from a line, etc. P: No they don't need to mean anything. Mathematicians are able to distinguish a group from a set simply by using a different "tag" or symbol to identify them. In one case they are identified using the symbol 'group', and in the other case they are identified using the symbol 'set'. For particular instances, statements such as "Let A be a set, and let G be a group" remove any ambiguity as to which is which. No appeal to meaning need be made in order to distinguish among sets, groups, or points, even though all of those terms are completely undefined and devoid of meaning.H: That's not really what I meant. I mean that if we reject even having some form of meaning of those terms then complete skepticism occurs in science and philosophy. P: Here again you are mixing up contexts. I agree that meaning must be a part of philosophy and science. It should only be "rejected" in the context of mathematics. Rejecting meaning from mathematics should not lead to skepticism in science or philosophy.>H: As for mathematics, I don't think you can construct mathematics without some meaning to those 'undefined' terms. P: You express a reasonable doubt. In fact, it is very difficult to do and was not even attempted until the past hundred years or so. Even now, people such as yourself and people studying pure mathematics for the first time, have a great difficulty in abandoning those cherished and deeply ingrained beliefs and ideas about such things as sets and points. But, it can be done, and it has been done. The resulting mathematics has much more power and potential than it did before.>>H: The reason that math is possible is because we can use the meaning of abstract terms and from that we can construct abstract definitions, equations, theorems, etc. P: Not true! The reason that math is possible is that human beings are capable of abstract thought. But the reason that math is *useful* is because, in non-mathematical contexts, we can ascribe meanings to abstract terms and from the equations and theorems of mathematics we can make remarkably successful predictions in the non-mathematical contexts.In fact, up until now, Dick himself has not accepted his paper as a pure mathematical exercise. I have tried to convince him that it is, but I have never put together a convincing case. I hope he will become convinced by the case I am making in this and my previous posts. As I said, I think that all three of us could come to an agreement if only we all saw his work as a theorem of mathematics.

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