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Impasse? -- Maybe Not. (Response To Harv)

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Posted by Paul R. Martin on March 3, 2002 18:07:50 UTC

Hi Harv,

First of all, thank you for your thoughtful response to me. Second, I apologize for being so slow to respond. Third, please don't let this thread distract you from the progress you and Dick are making.

It seems clear to me where you are having difficulties understanding what Dick has done. I think the problem is in the way you view the nature of mathematics. If I am right, then maybe I can help a little. If I am wrong, then I hope either you or Dick will set me straight.

***I know we've gone through this disagreement [of considering concepts devoid of meaning] before. I can only think we are not communicating since I cannot understand how you could believe that a meaningless concept is entertained by scientist or mathematician.***

You are right: we are not communicating. We can cut the problem in half, though. I will concede that scientists do not entertain meaningless concepts, nor do ordinary people. They are interested only in concepts that have meaning in our world. But the mathematicians,.....that's a different story. That is where we disagree.

***[Children] are taught to count objects (people, pens, dogs, etc). Only after all these tangible things are related to in a mathematical manner, do we finally leave the tangible things and just focus on the abstract things. After many years of all abstract concepts someone might feel that all these abstract concepts are 'meaningless' in terms of a tangible world, when in reality there is simply a distant connection of learning that it just appears that these abstract concepts have no meaning. When in fact, we are quick to connect those concepts to the tangible world if we are so lucky as to teach young children.***

One of the most shocking things I learned in my study of mathematics is that there are two completely different concepts of mathematics held by people. These are called "Pure" and "Applied" Math. The common one, Applied Math, is what we teach children. It involves counting things, doing calculations, and solving problems in the real world. The less common concept of mathematics, is the one held by mathematicians themselves, and it has nothing to do with counting things, doing calculations, or solving problems.

It is true that mathematicians may balance their checkbooks and do other calculations, and it is true that some of their mathematical work helps people do calculations or solve problems. But in doing Pure Math, those things are not the objective or even the subject. The objective of Pure Math is to construct a system of definitions, axioms, and theorems that are consistent with the rules of logic. In this process, great pains are taken to avoid and disclaim any relationship between the terms defined and anything in the "real world".

Now, if you understand and agree with that, then I think our disagreement boils down to the question of how successful can mathematicians be in eliminating all meaning from their terms.

You have pointed out, correctly, that over the history of mathematical development, the structure was built up on the foundation of empirical knowledge of the world. I will concede that. But, beginning about a century ago, mathematicians began deliberately divorcing their concepts from any connection to the world. Of course the concepts had "meaning" in the context of the logical mathematical structure, and if one liked, one could make a correlation between those concepts and something "real", but such a correlation played and plays no part in the mathematics.

You can suspect that mathematicians are kidding themselves about divorcing meaning from their concepts, but I am sure that if you took a course in the foundations of mathematics, you would discover that it can be, and has been, done. The major significant step in that work was when the Peano Axioms were developed from a set theoretic basis. That means that all of analysis (the mathematics of numbers) is based on the undefined concepts of sets, elements, and belonging.

***I cannot understand how you could believe that a meaningless concept is entertained by . . .[a] mathematician.***

Let me give you an example. I ask you to entertain the following imperative sentence:

'Let A be a set.'

That sentence, of course, is typical of the sentences entertained by mathematicians. Now, let's examine what meaning, if any, that sentence has. And, at the outset, I will agree that it has some meaning. But, the meaning has only to do with a communication between two minds and nothing to do with anything "outside" of those minds.

The meaning of the sentence is that it is a request, from one mind to another, that they agree to assign the symbol A to an instance of a concept that they agreed to identify as a set. (I used the third person pronoun because the two minds involved are the writer of the sentence and the reader of the sentence. In one specific case, I am the writer, and you, Harv, are the reader. But there may be some other unidentified readers, and in general, the writer's identity may be unknown.)

Now, we can agree that these two minds are part of the world, so the sentence does really have something to do with the real world. Okay, I'll agree with that. But that is not the issue. The mathematical content of that sentence is not concerned with the minds that are pondering it. The mathematical content of the sentence involves the symbol A and the concept of a set.

So do those things have meaning? Well, let's start with the symbol, A. Again we are back to the two minds. Symbols are the medium of communication that we humans customarily use. (There are claims of non-symbolic communication taking place, but I think scientists and mathematicians alike would disallow any telepathic or other non-symbolic communication in any of their work.) (N.B. all symbols used in communication can be reduced to, or encoded in, numbers.)

So, except for the fact that the symbol A has been agreed to by the two minds to signify a set, I ask you, Does A have any relationship to anything "real"? I think not. Even if you imagine that it does, and in your mind A represents a gaggle of geese, the mathematicians would adamantly claim that there is no relationship between A and your geese or anything else in the world. At least there is no such relationship that plays any part in the reason the writer has asked you to "Let A be a set".

Similarly the concept of a set is left undefined and thus completely disconnected from anything in the world, even though you most certainly conjure up mental images of examples of sets when you think about the term.

So I would agree with you that there is a cloud of meaning swarming all over all mathematical structures. But I hope you can see that mathematicians can entertain these concepts and still divorce their manipulation of them from any meaning having to do with any "external world".

***I think you are actually meaning that abstract concepts have no tangible meaning.***

Yes, if by "tangible" you mean having some correspondence with things in the world outside of the communication between two minds.

***If that is what you mean, then I agree only to a point. The meaning of an abstract concept (e.g., a mathematical point) is meaningful but in a different sense than a physical point is meaningful. The 'physical point' is not really a point (in the mathematical sense) but rather is only analogous to the way humans conceive of points. The issue is rather reversed since mathematical points actually come from our abstracting from 'physical points'. That is, we look at only certain qualities of a 'physical point' and we pay no attention to the elements which distract from our concept of a mathematical point. Eventually with enough mathematical learning, mathematical concepts become so abstract that they no longer hold any tangible meaning in the formal mathematical sense. This is what I believe you are talking about, right?***

No. That is not what I am talking about. What you described is the historical development of mathematics for the millennia leading up to the last century, and you are right. That is the way it happened. But what you described is not the way mathematical concepts are considered by modern mathematicians. Now, a concerted effort is made to make sure that any vestige of meaning, that might still taint the terms and concepts used, does not play any part in the chain of inferences that lead to the development of the mathematical theory. Mathematicians are keen to jump on any breach of this requirement if it should happen to slip into the development.

***Well, I am not concerned about the formal mathematical sense of meaning. I am talking about the epistemological sense of meaning.***

That's the key. We really don't have a disagreement at all. We are simply concerned about two different things. I am only concerned about the formal mathematical sense of meaning and not at all about the epistemological sense of meaning.

If I could somehow get your interest and get you to be concerned about formal mathematics for just one (this) paragraph, I would point out that the theory of mathematical analysis (i.e. numbers) can be built up from concepts that are completely devoid of any relationship with anything that might exist in any objective reality. With that understood, I would go on to point out that on that basis, Dick has proved that there are necessary constraints on any conceivable, or even possible, subsets of sets of numbers. Further, Dick points out that all symbols can be encoded in numbers, so any communication that is mediated by symbols can be reduced to a set of numbers. In other words, anything that can be described can be described in numbers. So it follows that anything that can be described in symbols must necessarily be constrained by the constraints Dick has discovered.

If you followed the previous paragraph, you will see that there is no dependency whatsoever on any "objective reality" or anything in any "objective reality" with the one exception of the minds that have developed, or come to understand, this logical structure and result.

Now, if you followed that, we can move on to epistemology.

***Epistemology is concerned about how we know what we know.***

Okay. But let's first look at "what we know" and then how we might come to know it. We can talk about what we know about the world, which is what I think you mean, but we can also talk about what we know about the mathematical and logical structure I just described which has no connection to the world. Those are two separate cases.

So, how do we know what we know? Well, let's take it case by case. I have just described the modern method of how we can come to know about the constraint Dick discovered. It can be, and has been, done without any appeal whatsoever to the world, except for the minds that contributed to the result. It is also true, moving on to the second case, that some of these constraints have been discovered empirically by scientists over the centuries by actually examining the world. This, of course, is the method most commonly used to learn what we know about our world.

Dick is simply offering an additional method for learning about the world. Think of it as an extension to epistemology. He has discovered that there are some things about the world that must necessarily be so as long as the world behaves consistently enough that its behavior can be communicated from one mind to another. It makes logical sense that if we started with his constraints before we began actually investigating the world by physical experiment and observation that we could maybe get a head start and also avoid some unnecessary work.

*** The epistemological issue becomes: how can mathematicians treat certain concepts as meaningful (i.e., useful to be discussed in mathematics) and yet place no tangible reality to those concepts. The answer, I believe, is that they place meaning on other abstract concepts and equations which 'sit on top' of a whole slew of other abstract concepts. At some point lower in this hierarchy (near the axioms) we see the tangible concepts that all the above is sitting upon.***

I don't believe that is the correct answer to your question. In fact, I don't think your question poses an important epistemological issue. In the first place, as I have tried to explain, mathematicians don't treat certain concepts as meaningful. It is the scientists and engineers, along with all the rest of us, who assign meaning to mathematical concepts.

Again, your description of axioms embodying tangible concepts is historically accurate. But that view no longer holds sway with mathematicians. They have succeeded in divesting their constructs from such meaning, believe it or not. If you don't believe me, ask some Pure Math professor and let me know what he/she says.

That's probably enough for now. Good talking with you, Harv. Sorry for the length of this post.

Warm regards,

Paul

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