It’s getting to be baby chipmunk season, and (oh, no!) the things get in the street and panic when they see a car coming. They run all over in a frenzy, including going off the road and back on, making it nerve-wracking to try to avoid hitting them. It is especially uncomfortable with a little girl in the car with you, (“Oh, little guy,” she cries). Or it is difficult for a young man trying to impress his date. (To paraphrase Oscar Wilde, she says, “Please don’t run over the rodent. He’s trying his best”).
Meanwhile, there is always a bird perched safely on a tree branch nearby, taking it all in with its avian brain how the mammalian brains aren’t quite figuring it out how to avoid colliding with one another. I suspect that the bird knows about such things (how the car is going to stay on the road, so don’t go back onto it once you have reached the side) because of having the additional perspective of the bird’s-eye view. From up there, the bird can see that a car moves by always staying on the road (whereas the rodent is stuck seeing things close-up from moment to moment without the benefit of the bigger picture).
And that brings me to my point. There seems to be a difference between just plain seeing a car coming at you and seeing a car coming at you that you know is going to stay on the road. And that seems to be a hint—or at least an analogy—for understanding some other significant issues in science and philosophy.
The physicist Frank Wilczek has discussed such matters—he calls it “big picture physics”—in his book A Beautiful Question: finding nature’s deep design. And the mathematician Michael Harris refers to something like that in “category mathematics,” introducing us to what he calls an “avatar perspective.” Wilczek tends to see the matter as an either/or issue (as in which is “right.” the bird or the rodent?), and he is interested in how that reflects different approaches in science. Harris, on the other hand, tends to format the issue as “the opposite of deduction” (in his book Mathematics without Apologies). And what does that mean, to use the opposite of deduction? Well, it isn’t induction. It is a technique, often compared to special relativity, for rendering the differences in perspectives beside the point (like in relativity how no view is privileged). So an avatar perspective is not really about either a bird’s-eye perspective, or a rodent’s, but about transcending the notion of perspective.
And it seems that a lot of difficult math proofs can be made by using that approach.
I have my own slightly different point I want to make about all this rodent business, but first I hope to answer that glaring question, “How is it really possible to eliminate the importance of perspective, especially without invoking absolutes?”
My interest is not to delve too deeply into the vagaries of category theory but just to answer that question (about perspective).
In “category theory” in mathematics, a number is defined by its relationships with other numbers (rather than being understood absolutely, as a thing-in-itself, regardless of anything else). But rather than pursue an endless regression of describing relationships between relationships and relationships between those relationships, we can appeal to a higher category and see a number in terms of that. But it is not a category as in Plato, where it works to deduce syllogistically from knowing a thing’s category. Rather, it is more like in Hinduism where a deity can “manifest” in different ways, and then a category is a grouping of all the ways that a number can manifest. So it ends up being a way of clustering all of those perspectives (all of those manifestations) without any single one of them being privileged compared to the others (at least for purposes of the math proof). Hence it is named an “avatar perspective.” Instead of employing equal signs representing equalities, the proofs use arrows representing the “morphisms” (as the numbers’ manifestations are called).
And that can be profitably contrasted with Bertrand Russell’s more traditional view of symbolic logic and the claims that are made for that. (Caution, however. I am not a Russell fan, not since I chose to write a term paper on him way back in my undergraduate days).
Russell went too far. Along with Whitehead, he is said to have proved that all of the formulations found in math (concepts such as greater than, and equivalence) are also in logic, so accordingly, math is said to be a subset of logic. (And so far, okay with me). But then, by some kind of slip-sliding along, Russell turned that into treating math and logic as basically interchangeable (not okay with me, since there are other ways of being logical besides with math. When the question on an exam in a chemistry lab states, “Describe the logic of this experiment,” it isn’t going to do just to cite your p’s and q’s).
In other words, just because a premise works in mathematics doesn’t automatically mean that it works in the rest of the physical world.
Russell, with his doctrine that he called Rationalism, claimed that many traditional problems in philosophy were better solved as problems in the logic of math, with many applications. (I have discussed how he applied that to causality) More generally, Rationalism as a philosophy holds that being rational means having a reliance on deduction and/or avoiding logical contradiction. But that is instead of relying first of all on sense data, as in Empiricism.
So it is in terms of comparing Rationalism versus Empiricism that an avatar perspective might prove enlightening (since recall that an avatar perspective is the opposite of deduction).
But first of all, regarding Russell:
The good thing about Russell—his saving grace, so to speak—was that he was always changing his mind. (His curious mind was always growing, we might say). He even once helpfully wrote a book titled “What I Believe Now” (or something like that) to enable his readers to stay up to date with him. And lo! He even came to change his mind regarding Rationalism (which was actually pretty big of him, because it came down to taking back a lot of what he had said throughout his career).
He ended up deciding that physics was “at war with itself” when it came to trying to use just his symbolic logic to describe the real physical world. The reasons could well have been the familiar ones, such as how in the real world there are degrees of being true or false (not all-or-none as in symbolic logic), and, in the real world, things do not break evenly into well-defined categories the way that numbers do. But what worried Russell was that logic cannot prove the existence of the world described by physics, so that we have to take that on faith (which is unscientific). And that undermines Rationalism and the notion that logic is more fundamental than sense data. (Of course, empiricists do not worry about deriving the existence of the world from pure logic because they just test, instead).
Then to complete the comparison with category theory: Harris does not mention it, but the way that perspective can be transcended is by becoming more abstract. Mathematicians call it “level shifting” when they appeal to a higher level of abstraction (meaning to remove details). And it is by becoming more abstract that category theory is “the opposite of deduction.” Mathematicians liken it to “twisting” a shape, as opposed to organizing everything into sets.
So category theory shows that math itself does not have to be about deduction, as is presumed in Rationalism. Empiricists know that math can also be used to describe patterns, for instance, and change, and maybe even changes such as causality. And morphisms. Logic can even be used to describe “becoming.” (Recall that category theory uses arrows).
Empiricist see relationships among things and describe those relationships using math. But that does not mean that the world is organized solely according to set theory and/or that mathematical logic prefigures the physical world.
So here is where I hope to announce my own take on all of these various approaches. Elsewhere I have been arguing for the philosophical position that the world seems full of lots of little logics rather than relying on just one Universal logic that applies to everything as in Russell’s Rationalism. The little logics are created from the many different ways in which things in the empirical world fit together. (That is how empiricism fits with logic; the way that setups become arranged creates the logic of what can happen under that scenario). For instance, in chemistry, if we add together two solutions, we get a different product depending on the temperature, air pressure, pH, amount of stirring, or what else is in the solution such as a catalyst. We do not always get one answer absolutely, regardless of anything else, but rather actions in the real world depend on context. Each outcome has its own logic of how events happen under that setup of circumstances.
And then “rationality” means being able to have facility with many different little logics all at once (rather than there being one big logic that applies to everything, as advocated by Russell). I suppose that it could be argued that seeing all the little logics altogether amounts to having an avatar perspective (in the sense of there not being any absolutely correct logic or perspective). But I am not saying that because I am not sure I would embrace all of the overtones implied by category theory. What I do believe is that most things in this world are relative, but absolutes are not really necessary for decision-making. Relativism should not mean that all options are equally as good as any other, or, that we cannot tell the difference between options without having an absolute to compare with. The way to handle relativity is by finding the logic that appears in how multiple relative events fit together. Putting things into an arrangement gives them utility, even when in isolation they lack dependable qualities. (I gave some extensive examples of that in my last post).
Another example of rationality is how kinetics can sometimes trump thermodynamics to change an outcome. There is a logic to kinetics (about how fast events occur), and there is a logic to thermodynamics (about how stable a result is). And the two can yield different overall outcomes. So my point is that it requires rationality to see the two logics together and to know which will prevail.
Who gets the girl? The boy who asks her out first, or the one who is the best most stable match for her?
To navigate the world, we have to see the multiple logics that exist in all the little ways that things can fit together.
Do we go through life like the baby chipmunk and experience each moment in isolation? Or is rationality the ability to see each moment in light of remembered others, so that we see the logic in how specific situations are fitting together, such as knowing that a car with stay on the road as we meet it while walking?