The foundation of mathematics
Then what is mathematics all about, basically? This is tied to the question of why it works, since in itself it is a closed system and neither needs verification nor application to justify itself. Nevertheless it does indeed apply to a wide range of things, and especially to what is measurable.
This is why many philosophers, from ancient times on, have thought that mathematics is the science of quantity. But, though I think its major application is in dealing with what is quantified, I don't think that this is really its essence.
As I see it, this is what mathematics is:
Mathematics is the science of relationships and the related as such.
Most of the relationships mathematics has so far concerned itself with have been quantitative ones; but contemporary mathematics has gone rather beyond this and begun exploring relationships that don't necessarily involve counting and measuring, such as the "belonging to" relationship of set theory. And as mathematics discovers more and more what it is doing, it is quite possible that there could be branches of mathematics that explore relations like causality, similarity, inherence, and so on; and who knows? Some of these might turn out to be fruitful and applicable to problems in the real world.
At any rate, the tie of mathematics to the real world is that in the real world, and certainly in the real world as known by us, there are relationships. It isn't surprising that an exploration of what a given one of these relationships is and what is implied in having things related in this way would have an application to things that are related in the way in question.
One of the reasons mathematics is difficult for ordinary people to follow is that we start from objects and abstract the relationships from them, as we saw in Section 3 of the third part. Mathematics supposes the relationship to have already been discovered, and doesn't care what it came from. We discover, for instance, that objects belong to classes of, say, similar objects. Mathematics says, "Let's look at what 'belonging to' means and implies."
Hence, 1. Mathematics starts with the relationship itself.
2. Mathematics then defines (i.e. makes up imaginary) "objects," whose sole meaning is "to be the object of this particular relationship."
3. Mathematics then asserts a set of basic facts about these invented "objects" based on the meaning of the relationship they have with each other. These are the "axioms" of the system.
4. Mathematics then draws out the logical implications of these facts about the objects related in this way. These are the "theorems" of the system.
One of the reasons formal logic doesn't work as a mathematical system is that statements have meaning as well as truth, and the two can't be divorced from one another. Hence, if you want to talk about "truth-functions" and create objects called "propositions" which are supposed to have nothing but truth or falsity and connectives which are supposed to be nothing but truth-functional, you are falsifying the relationships of statements with each other and with each other's truth; and so your logic will not work.
On the other hand, you can separate out "belonging to" or "in addition to" or "beside" from other relations the objects have; and so there is no falsification going on if you explore just the relationship, say, of "belonging to" by making up objects called "sets" which you then define as "what is belonged to" and "members" whose definition is "what belongs to."
And right here is where most people who aren't of a mathematical turn of mind have one of their major difficulties. "Yes, but what is a set?" they ask. "Give me an example of one." If a mathematician is being true to his calling, he precisely can't give an example of a set, because there's nothing in the world that does absolutely nothing but get belonged to by members--or better, does absolutely nothing but get involved in belonging-to relations, since sets can belong to other sets. But they can't do anything else but belong to or be belonged to; because they were invented precisely to do nothing else, so that the relationship "belonging to" could be explored without any distraction.
And that's the idea of these "objects." Since they have no other raison d' être than to be "whatever is related by the relation in question," and therefore they have only the one aspect which is the foundation of this relationship and no other, then the dream of the empirical scientist is fulfilled: to have isolated the objects of his investigation from all distracting acts and characteristics.
So, exactly backwards from the way we normally understand, with the objects and their messy multiple acts first, and the relations understood from them, mathematics takes the relationship first and derives the objects from it. In so doing, of course, it sacrifices having its objects tied to the real world; only the relationship itself has any tie to the real world. But of course, since there are real objects related in this way, then the mathematical objects will be abstractions of them.
But it is important to see that the mathematician does not get his objects by abstracting from real objects that have the relationship he is interested in and finding what is "common" among all of them. This would make his science empirically verifiable or falsifiable, and it isn't. No, the objects are simply made up because relations need relata, and these relata are created to be nothing but the relata of this relation. Thus, when a mathematician "defines" a set as "a collection of objects," he is just helping you out and kowtowing to your way of thinking, so you won't put him away in a padded cell. He knows that this isn't what a set is, because "collections" have all sorts of properties in addition to "being belonged to," and "objects" have more to them than just belonging to sets. It is only when you get fairly deeply into mathematics that he lets you in on the secret--and probably not even as explicitly as I have done. Certainly none of my mathematics teachers ever did, and I have studied some mathematics at the postgraduate level. I've had plenty of hints, but no one came right out and said it in so many words.
Now of course, the mathematical system in question will only be applicable to real objects insofar as the mathematician's objects have the single property which is one property of real objects related in the same way; but if they don't, his system isn't false, but just inapplicable to the real world. Actually, the inapplicability wouldn't be the fault of the objects, exactly, but that the relation in the real world is different from what the mathematician meant when he used the same word. Thus, for instance, set theory does not fit the relation of "belonging to" in the sense of ownership; neither a member nor a set owns anything.
But mathematicians like to justify their existence as much as anyone else does, and so they try to make the relations they are dealing with as nearly as possible the same as the basic meaning (or at least one common meaning) of the word they use to express it. To the extent that they succeed, to that extent the mathematical objects will be abstractions of the real objects related in this way, and the math will apply.
Now the relation is defined in mathematics by the axioms. The "definitions" deal with what the objects are in terms of what the relation is; the relation itself is defined as a series of facts of what these objects do to each other (i.e. how they connect with each other). Thus, for instance, one of the axioms of set theory is that a member belongs to a set, but a set may not belong to a member. Another is that a set may belong to another set, in which case it is called a "subset." Another is that if a set is a subset of another set, all of its members are also members of the other set, and so on.
The idea here is to get as few statements as possible that define the relationship exactly and give all and only the independent possibilities and the impossibilities for the objects of that particular relationship. These are the axioms.
It is easy enough to make up a set of axioms for some relationship you just create out of whole cloth and give a name to, like the relationship of "jonesing." You define objects such as smiths and knopfs and then make up axioms like, "Every smith can jones a knopf, but two and only two knopfs can jones a smith." "If a smith joneses a smith, then it cannot simultaneously jones a knopf." "A knopf can jones one knopf, but the result is two knopfs." And so on.
What are you talking about? That's it. Precisely nothing, because in the system there is no meaning to "jonesing" except what the axioms say--and of course no meaning to "smiths" and "knopfs" except that they "jones" each other according to the axioms. In this case "to jones" has no meaning outside the system either, so it's all a game; it's when the relationship means something outside the system that mathematics makes sense to non-mathematicians--and mathematicians can hope to get paid for doing their thing.
The tricky part of the axioms comes when you're dealing with a relationship that means something in the real world. Then you have to see to it that (a) your axioms exhaust all of the independent aspects of this relationship, (b) that you don't introduce something as an axiom that only sometimes is true of the relationship in the real world; and (c) that you don't bring in something that seems to be part of the relationship in the real world but is actually a different property of the objects that happen to be related in this way.
Euclid, who was a towering genius, introduced into his geometry the famous "parallel postulate" (axioms and postulates nowadays mean the same thing), that one and only one line parallel to (i.e. never meeting) another line can be drawn through a point. It turns out that this is true only of lines on what we normally think of as a flat surface (plane geometry); but on other types of surfaces, it doesn't apply at all. Hence, it isn't an axiom of geometry as such, but only of one specific type of geometry, not surprisingly called, "Euclidian geometry." The point here is that if Euclid couldn't spot what was irrelevant to what he was doing, we lesser lights are going to have a much worse time picking out all the axioms and ensuring that we have only the axioms for any applicable branch of mathematics.
What happens next is that these different possibilities are combined in various ways to generate new statements about the objects based on the axioms; and these are the theorems.
Beyond that, there is the use of the particular mathematical system by those who want to apply it to the objects that are related in the way in question. These people are not interested in proving theorems from the axioms, but in making statements and drawing out implications from them.Next