***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.***
They aren't eliminating meaning, they are eliminating tangibility. That's a totally different story. Meaning is still there, but it is an abstract meaning (versus a tangible meaning).
***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.***
Just to clarify, you acknowledge that mathematical terms have meaning within the context of their mathematical structures, correct?
***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.***
By having 'meaning' I am not saying they have tangible properties. I am merely saying that they have structural properties that relates to the structure that gives those terms certain types of meaning. All mathematical terms have meaning - albeit abstract meaning. Can we agree?
***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.***
They are undefined in terms of non-essential characteristics. For example, elements are structures within a set, yet an element is left undefined. It simply isn't essential for an element to be defined as something, but elements are understood by set theorists as to compose a set.
***H: I cannot understand how you could believe that a meaningless concept is entertained by . . .[a] mathematician. P: Let me give you an example... 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.***
Exactly. The meaning is abstract. But, there is implied meaning (otherwise it would be gibberish).
***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".***
I agree that A has an abstract meaning lacking tangible reference to geese or anything for that matter.
***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.***
The connection is there originally and we learn how to abstract away from those tangible things to learn to deal mathematically with sets, points, numbers, etc. If sets were completely disconnected from things in the world, then there would be no way to learn about sets. How do you think humans came to identify sets?
***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".***
That's fine. If you want to work in set theory, it doesn't require us to consider how we know of sets or how the history of mathematics.
***H: 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? P: 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.***
I realize that mathematicians are eliminating these tangible connections with their concepts. Mathematically they are successful. Epistemologically they are not. They leave a trail (call it an epistemological trail). This epistemological trail is a smoking gun that says that all mathematical concepts have a causal relationship with sensory things. What this says is that no knowledge is 'free' from what came before. You can escape the past mentally (i.e., in the present moment), but you cannot escape from the casual link of the past.
In terms of epistemology, the concern is the causal link. If there exists a causal link, then any concept is not just what it is today, it is a sum of all events that occurred that allowed you to reach that point of maturity. Our mathematical knowledge must always stand on the shoulders of giants.
***H: Well, I am not concerned about the formal mathematical sense of meaning. I am talking about the epistemological sense of meaning. P: 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.***
A line has been crossed here that I feel is important to draw attention. We are concerned not about some mathematical problem, but we are concerned about the representation of reality. If Dick's work was limited to describing mathematical objects in terms of numbers (e.g., translating topological designs into numbers), then his thesis should be considered a mathematical work having nothing apparent to do with physics or science. But, this is not where Dick stops. He wishes to say something about reality and how reality can be represented. This is where that 'epistemological trail' becomes a significant issue.
***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.***
The dependence is that mathematical concepts are linked to sensory experience (via that epistemological trail that I mentioned).
***H: Epistemology is concerned about how we know what we know. P: (...) 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.***
I see unsolvable problems using this approach by Dick. Let me mention a few:
1. Does Dick's method generate new knowledge of the world?: So far, I haven't seen any new predictions. If it doesn't generate new knowledge, then how could it be a new approach to of anything significant? It would be like finding an Egyptian fortune teller's book with excellent confirmed predictions that followed the next millenium (i.e., all the predictions are confirmed for a 1,000 years after it was written). The bad news is that we don't for sure know when it was written nor do we have any future predictions by which to test it. What good is it to anyone but Egyptologists?
2. How to validate the defined concepts?: For example, Dick's definition of time is somewhat contrived (in my opinion). It is based on subjective (conscious). How does time exist prior to conscious beings? You have to restate the definition in terms of 'if there were conscious beings present then time would be...'. This is not acceptable for an epistemological approach. Further, how do we know that Dick simply didn't select his concepts so that logically he could lay a basis for the mathematical equations that were to come? How do we know that all of this isn't some house of cards where he simply set the conditions to emulate the equations of QM and classical physics? Monday morning quarterback's would make a fortune if their services could be used on Sunday, but unfortunately by then it is too late. Dick's model came after the fact and the only reason that he felt he was successful was because he obtained equations made famous by science prior to his efforts. Epistemologically, this is all wrong in terms of substantiating a new means to acquire knowledge.
3. 'Where do the laws of math and physics come from'? It is not an answerable question (at least in 2002). The laws of physics might come from our limitations at understanding nature, or they might 'exist'. Mathematics is the same situation (except it might be a limitation in our thought processes versus a limitation of our understanding nature). For that reason, we can't say that Dick's model does any epistemological service to science. Afterall, if mathematics is an invention, then so is Dick's model. It doesn't exist, it is merely a clever human invention.
***H: 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. P: 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.***
I think you acknowledged that those concepts have meaning within the mathematical structures they are used, right?
***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.***
I'm talking about the epistemological trail. This is what impacts Dick's assumptions.
Warm regards, Harv