there's no place to start. We fear that when some meanings form such a circle, then there would be no way to break into the circle, and everything would be too subjective to make good science.
I don't think that we should fear the fact that our meanings and definitions run around in vicious circles, each depending on the others. There's still a scientific way to deal with this: just start making new kinds of theories about those circles themselves! You don't have to break into them you only need to have good theories about them. Of course, this is hard to do, and likely to get complicated. It was to avoid complication that all those old theories tried to suppress the ways that meanings depend on one another. The trouble is, that lost all the power and the richness of our wondrous meaning-webs! Let's face another fact: our minds really are complicated. perhaps more so than any other structure Science ever contemplated. So we can't expect the old ideas to solve all the new problems.
Besides, speaking of breaking into the meaning-circle, many science-fiction writers have pointed out that no one ever really wants to get oneself inside another mind. No matter if that's the only hope of perfect communication of being absolutely sure you understand exactly, at every level of nuance what other people mean. The only way you could do that is by becoming exactly like that person but even then the game is lost, since then you couldn't understand any more (perfectly, that is) just what it was that your old self had tried to say.
What Is a Number, That a Mind Might Know It?
Now let's return to what numbers mean. This time, to make things easier, we'll think about Three. What could we mean by saying that Three hasn't any single, basic definition, but is a web of different processes that depend upon each other? Well. consider all the roles 'Three" plays.
One way a person tells when there's a Three is to recite "One, Two, Three." while pointing to the different things. Of course. while doing that, you have to manage to (i) touch each thing once and (ii) not touch any twice. One easy way to do that is, to pick up one object, as you say each counting word, and remove it. Soon, children learn to do that in their minds or, when it's too hard to keep track, to use some physical technique like finger-pointing.
Another way to tell a Three is to establish some Standard Set of Three things. Then you bring your set of things there and match them one-to-one: if all are matched and you have nothing left, then you had Three. And, again, that "standard Three" need not be physical; those three words, "One, Two, Three" would work quite well. To be sure, this might make it hard to tell which method you're using "counting" or "matching'' at the moment. Good. It really doesn't matter, does it'' (Except, perhaps, to philosophers.) For do-ers, it's really good to be able to shift and slip from one skill-process to another without even realizing it.
Another way to know a Three is by perceptual groups. One might think of Three in terms of arranging some objects into groups of One and Two. This, too, you can do mentally, without actually moving the objects, or you might lay them out on a table. You might learn several different such arrangements:
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For Five you have more families of ways. because you can use groups of Two and Three, or groups of One and Four. A pentagon, a thing-filled square, a W, a star, a plane, a cup; they all make Fives.
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Which way is right to count, or match, or group which is the "real" meaning of a number? The very question shows how foolish is any such idea: each structure and its processes have both their own uses, and ways to support the others. This is what takes the whole into a powerful. versatile skill-system. Neither chicken nor egg need come first; they both evolve from something else.
It's too bad that so many scientists and philosophers despise such networks and only seek to construct simple "chains" of definitions in which each new thing depends only on other things that have been previously defined. That is what has given "reductionism" a bad name. The common sense meaning of Three is not a single link in one long chain of definitions in the mind. Instead, we simply let the word activate some rather messy web of different ways to deal with Threes of things, to use them, to remember them, to compare them, and so forth. The result of this is great for solving problems since, when you get stuck with one sense of meaning, there are many other things to try and do. If