A number of years ago I got into a discussion with an acquaintance about what kind of symbol system tells us “the truth” about the world. This is not how my interlocutor expressed the problem; she simply insisted that mathematics gives us the truth. I tried many different approaches to get her to understand that what she was saying made absolutely no sense, because the first thing that must happen (once any collection of symbols is at hand) in order to talk about truth was that those symbols have to be interpreted, and such interpretation is not given in advance. Thus, I have a modest background in some advanced forms of mathematics (mainly formal logic, abstract algebra, and a touch of differential geometry), and I understand that simply having a bunch of squiggles in front of you is not enough to adjudicate whether those squiggles say anything at all, much less anything that is true. Meanings must be assigned to those squiggles such that they hang together to form some kind of model, and that model then must be associated with the world in some form such that the model can be interpreted as making claims about the world which then can be interpreted as to its truth content. And here, “world” can mean either the world of concrete experience or a purely abstract “world” which is itself something of a mathematical construct. Also, my choice of the term “truth content” rather than “truth value” is not an innocent one: I wish to leave open the possibility that truth evaluations can be more complex and multi-dimensional than the mere assignment of values.
It became very clear that while I understood my acquaintance’s position, she in no way understood mine. This was because while I was repeatedly able to paraphrase – that is, interpret – her argument, when asked to do the same for mine she was unable to do anything other than repeat her own position, which addressed none of the points I had made. In later years, she was known to crow a bit about how she “won” the argument. To be fair, in retrospect I realize that there were a number of ways I could have made my own position clearer, as it was burdened by a much greater degree of philosophical nuance than the position she was presenting. And I confess that I do not think quickly on my feet; indeed, I’ve only ever suggested that, given time, I can think thoroughly. (One of the reasons I went into philosophy is because a line like, “Herstein! If we don’t get this metaphysical principle out the door by end of business today, our competition is going to crucify us!” is not something one is ever likely to hear from one’s department head.)
Let us start with something simple: Most all of us will immediately assent to the statement that
“1 + 1 = 2” is true.
But there are, in fact, already significant problems here. Were we to show the squiggles between the quotation marks to aboriginal persons who’d never had any contact with the west, they would have no idea what it was they were being shown. (Is it abstract art? Religious iconography?) But even for those of us well acquainted with the symbolism, we cannot assert the truth of the above without qualification. Thus, one quart of water plus one quart of alcohol does not equal two quarts of fluid; similarly, one bucket of fine sand plus one bucket of course pebbles does not equal two buckets of material. But even demanding a high degree of uniformity in the things counted will not save us in every situation. An atom of uranium added to a second atom of uranium won’t equal two atoms, if one of them decays to lead in the aftermath of the addition. And the criteria of uniformity might be ones that are not at all obvious to us. Our aboriginal peoples might, after explanation, understand our general rule of addition, and still insist that one sheep plus one sheep does not equal two sheep. (One of the sheep may be young and fertile, the other old and good for little more than mutton.) The only place where the rule “1 + 1 = 2” works out in general is in contexts that have been rendered rigidly uniform and abstract. That process of rendering is the stipulation of the valid range of interpretation for our symbols. Only within that range does “1 + 1 = 2” say something that can be legitimately characterized as “true.” The situation only gets worse as one moves into more advanced areas of mathematics. For example, Whitehead appreciated the tensors Einstein employed in the development of the general theory of relativity, but objected in the strongest terms to their geometrical interpretation (which Einstein thought to be their most important feature.)
What does the word “earth” mean? For many people, especially science geeks, the first (and, generally only) meaning that comes to mind is an oblate spheroid orbiting approximately one Astronomical Unit from the sun. For someone who insists that “mathematics is truth,” that will likely be the only meaning to which the word “truth” can be applied …
… just like 1 + 1 always and only equals 2 …
But to a gardener or a farmer, it is likely to mean soil, and soil of a richer, more fertile sort than mere dirt. And this scarcely exhausts the possibilities. Consider, for example, what the Hopi people of North Eastern Arizona mean, when they speak of “the Earth.”
A visitor to the Thee Mesas area of Arizona might not be immediately struck by the thought of this particular land as being a vibrant source of life, but that is how the Hopi view it. They have a complex and not entirely uniform (across the Three Mesas area) mythology of creation, salvation, and restoration. The Hopi are a very private people, and many anthropologists suspect that such stories as have been shared may be at least partially no more than fodder provided to outsiders. (Early studies of the Hopi may be found HERE.) What we can be certain of, however, is that the word “earth” does not, for the Hopi, refer to an oblate spheroid orbiting approximately one Astronomical Unit from the sun. Does that suffice to show that what they are saying is false or wrong?
Well, it scarcely deserves mentioning that in whatever way(s) Hopi religious and spiritual symbolisms of the Earth map onto the reality at large, it will not be in the same way that statements like “earth is an oblate spheroid orbiting approximately one Astronomical Unit from the sun,” or “1 + 1 = 2” do so. In finding and creating what might be called “the truth of their lives,” the Hopi may well find those last two much more indicative of the coming wave of exploitive destructiveness which the Hopi stories speak of, in which the Hopi will be saved once again by the “Ant People,” rather than anything that connects meaningfully with The Earth. Only when outsiders try to forcibly interpret these stories as literal representations of natural history do we righteously imagine ourselves in a position to dismiss them as “mere” stories.
And the real problem with such an interpretation is not with the natural history part, but the presumptuous charge of literalness. Indeed, the idea that such a thing as a “literal interpretation” of anything is even possible, much less actual, is one of the more pathological pieces of intellectual rot to have ever infested western thinking. The putrid notion of literalness assumes that a model can only be interpreted one way, and that way is not only the only correct one, but is ideally simple in its direct, one-to-one connections between the model and reality. But this assumes that reality itself so catastrophically simple as to be entirely trivial. It is only through metaphor and analogy that we find such broadly pragmatic forms of relatedness that we get any leverage at all on a world that is vastly more subtle than our wits for conceiving it. Even (and even especially) Science and mathematics are shot-through with metaphors and analogies. Consider: a Hilbert “space”? A hypercomplex “number”? Only by washing these mathematical structures through multiple layers of abstraction do they analogically take on the labels of “space” or “number.” (The same can also be said of “gravitational waves.”)
People like Pat Robertson or Richard Dawkins lack the wit to understand how models or interpretations work, and so get into loggerheads over their simplistic forms of literalism, making things like religious wars all but inevitable. Faced with such intransigent and willful ignorance, one can only wish that the “Ant People” will be generous, and save others than just the Hopi.
” And here, “world” can mean either the world of concrete experience or a purely abstract “world” which is itself something of a mathematical construct.”
What is “a purely abstract world”?
If by “world” you mean “the world of concrete experience”, then the question is whose concrete experience?
Gary Herstein said:
Per your two questions, your first is largely answered (via example) in the text you quoted. Thus a mathematical model (as in model theory) is “interpreted” by a collection of equally abstract mathematical objects. I would add that, in the context of abstraction, “world” is scare quoted.
Your second question goes rather further astray, as it misses the point (which, I confess, I thought was fairly clear) that I was going no further than establishing a spectrum across the possibilities of interpretation. There is no purpose to be served by drilling into particular details that serve no purpose in illuminating the general point. (Here, also, I use “particular” and “general” in their logical senses to qualify my overall point.)