30 Some Historical (and Philosophical) Background

The human activity that we now call “mathematics” can presumably trace its origins into prehistory. What might have started as “a single goat”, “a pair of goats”, etc. became a story of abstract numbers that could be indicated purely by things like tally marks. In Babylonian times the practicalities of a city-based society led to all sorts of calculations involving arithmetic and geometryand basically everything we now call “mathematics” can ultimately be thought of as a generalization of these ideas.

The tradition of philosophy that emerged in Greek times saw mathematics as a kind of reasoning. But while much of arithmetic (apart from issues of infinity and infinitesimals) could be thought of in explicit calculational ways, precise geometry immediately required an idealizationspecifically the concept of a point having no extent, or equivalently, the continuity of space. And in an effort to reason on top of this idealization, there emerged the idea of defining axioms and making abstract deductions from them.

But what kind of a thing actually was mathematics? Plato talked about things we sense in the external world, and things we conceptualize in our internal thoughts. But he considered mathematics to be at its core an example of a third kind of thing: something from an abstract world of ideal forms. And with our current thinking, there is an immediate resonance between this concept of ideal forms and the concept of the ruliad.

But for most of the past two millennia of the actual development of mathematics, questions about what it ultimately was lay in the background. An important step was taken in the late 1600s when Newton and others “mathematicized” mechanics, at first presenting what they did in the form of axioms similar to Euclid’s. Through the 1700s mathematics as a practical field was viewed as some kind of precise idealization of features of the worldthough with an increasingly elaborate tower of formal derivations constructed in it. Philosophy, meanwhile, typically viewed mathematicslike logicmostly as an example of a system in which there was a formal process of derivation with a “necessary” structure not requiring reference to the real world.

But in the first half of the 1800s there arose several examples of systems where axiomswhile inspired by features of the worldultimately seemed to be “just invented” (e.g. group theory, curved space, quaternions, Boolean algebra, ...). A push towards increasing rigor (especially for calculus and the nature of real numbers) led to more focus on axiomatization and formalizationwhich was still further emphasized by the appearance of a few non-constructive “purely formal” proofs.

But if mathematics was to be formalized, what should its underlying primitives be? One obvious choice seemed to be logic, which had originally been developed by Aristotle as a kind of catalog of human arguments, but two thousand years later felt basic and inevitable. And so it was that Frege, followed by Whitehead and Russell, tried to start “constructing mathematics” from “pure logic” (along with set theory). Logic was in a sense a rather low-level “machine code”, and it took hundreds of pages of unreadable (if impressive-looking) “code” for Whitehead and Russell, in their 1910 Principia Mathematica, to get to 1+1=2.

 

    Courtesy of archive.org

Meanwhile, starting around 1900, Hilbert took a slightly different path, essentially representing everything with what we would now call symbolic expressions, and setting up axioms as relations between these. But what axioms should be used? Hilbert seemed to feel that the core of mathematics lay not in any “external meaning” but in the pure formal structure built up from whatever axioms were used. And he imagined that somehow all the truths of mathematics could be “mechanically derived” from axioms, a bit, as he said in a certain resonance with our current views, like the “great calculating machine, Nature” does it for physics.

Not all mathematicians, however, bought into this “formalist” view of what mathematics is. And in 1931 Gödel managed to prove from inside the formal axiom system traditionally used for arithmetic that this system had a fundamental incompleteness that prevented it from ever having anything to say about certain mathematical statements. But Gödel seems to have maintained a more Platonic belief about mathematics: that even though the axiomatic method falls short, the truths of mathematics are in some sense still “all there”, and it’s potentially possible for the human mind to have “direct access” to them. And while this is not quite the same as our picture of the mathematical observer accessing the ruliad, there’s again some definite resonance here.

But, OK, so how has mathematics actually conducted itself over the past century? Typically there’s at least lip service paid to the idea that there are “axioms underneath”usually assumed to be those from set theory. There’s been significant emphasis placed on the idea of formal deduction and proofbut not so much in terms of formally building up from axioms as in terms of giving narrative expositions that help humans understand why some theorem might follow from other things they know.

There’s been a field of “mathematical logic” concerned with using mathematics-like methods to explore mathematics-like aspects of formal axiomatic systems. But (at least until very recently) there’s been rather little interaction between this and the “mainstream” study of mathematics. And for example phenomena like undecidability that are central to mathematical logic have seemed rather remote from typical pure mathematicseven though many actual long-unsolved problems in mathematics do seem likely to run into it.

But even if formal axiomatization may have been something of a sideshow for mathematics, its ideas have brought us what is without much doubt the single most important intellectual breakthrough of the twentieth century: the abstract concept of computation. And what’s now become clear is that computation is in some fundamental sense much more general than mathematics.

At a philosophical level one can view the ruliad as containing all computation. But mathematics (at least as it’s done by humans) is defined by what a “mathematical observer like us” samples and perceives in the ruliad.

The most common “core workflow” for mathematicians doing pure mathematics is first to imagine what might be true (usually through a process of intuition that feels a bit like making “direct access to the truths of mathematics”)and then to “work backwards” to try to construct a proof. As a practical matter, though, the vast majority of “mathematics done in the world” doesn’t follow this workflow, and instead just “runs forward”doing computation. And there’s no reason for at least the innards of that computation to have any “humanized character” to it; it can just involve the raw processes of computation.

But the traditional pure mathematics workflow in effect depends on using “human-level” steps. Or if, as we described earlier, we think of low-level axiomatic operations as being like molecular dynamics, then it involves operating at a “fluid dynamics” level.

A century ago efforts to “globally understand mathematics” centered on trying to find common axiomatic foundations for everything. But as different areas of mathematics were explored (and particularly ones like algebraic topology that cut across existing disciplines) it began to seem as if there might also be “top-down” commonalities in mathematics, in effect directly at the “fluid dynamics” level. And within the last few decades, it’s become increasingly common to use ideas from category theory as a general framework for thinking about mathematics at a high level.

But there’s also been an effort to progressively build upas an abstract matterformal “higher category theory”. A notable feature of this has been the appearance of connections to both geometry and mathematical logicand for us a connection to the ruliad and its features.

The success of category theory has led in the past decade or so to interest in other high-level structural approaches to mathematics. A notable example is homotopy type theory. The basic concept is to characterize mathematical objects not by using axioms to describe properties they should have, but instead to use “types” to say “what the objects are” (for example, “mapping from reals to integers”). Such type theory has the feature that it tends to look much more “immediately computational” than traditional mathematical structures and notationas well as making explicit proofs and other metamathematical concepts. And in fact questions about types and their equivalences wind up being very much like the questions we’ve discussed for the multiway systems we’re using as metamodels for mathematics.

Homotopy type theory can itself be set up as a formal axiomatic systembut with axioms that include what amount to metamathematical statements. A key example is the univalence axiom which essentially states that things that are equivalent can be treated as the same. And now from our point of view here we can see this being essentially a statement of metamathematical coarse grainingand a piece of defining what should be considered “mathematics” on the basis of properties assumed for a mathematical observer.

When Plato introduced ideal forms and their distinction from the external and internal world the understanding of even the fundamental concept of computationlet alone multicomputation and the ruliadwas still more than two millennia in the future. But now our picture is that everything can in a sense be viewed as part of the world of ideal forms that is the ruliadand that not only mathematics but also physical reality are in effect just manifestations of these ideal forms.

But a crucial aspect is how we sample the “ideal forms” of the ruliad. And this is where the “contingent facts” about us as human “observers” enter. The formal axiomatic view of mathematics can be viewed as providing one kind of low-level description of the ruliad. But the point is that this description isn’t aligned with what observers like us perceiveor with what we will successfully be able to view as human-level mathematics.

A century ago there was a movement to take mathematics (as well, as it happens, as other fields) beyond its origins in what amount to human perceptions of the world. But what we now see is that while there is an underlying “world of ideal forms” embodied in the ruliad that has nothing to do with us humans, mathematics as we humans do it must be associated with the particular sampling we make of that underlying structure.

And it’s not as if we get to pick that sampling “at will”; the sampling we do is the result of fundamental features of us as humans. And an important point is that those fundamental features determine our characteristics both as mathematical observers and as physical observers. And this fact leads to a deep connection between our experience of physics and our definition of mathematics.

Mathematics historically began as a formal idealization of our human perception of the physical world. Along the way, though, it began to think of itself as a more purely abstract pursuit, separated from both human perception and the physical world. But now, with the general idea of computation, and more specifically with the concept of the ruliad, we can in a sense see what the limit of such abstraction would be. And interesting though it is, what we’re now discovering is that it’s not the thing we call mathematics. And instead, what we call mathematics is something that is subtly but deeply determined by general features of human perceptionin fact, essentially the same features that also determine our perception of the physical world.

The intellectual foundations and justification are different now. But in a sense our view of mathematics has come full circle. And we can now see that mathematics is in fact deeply connected to the physical world and our particular perception of it. And we as humans can do what we call mathematics for basically the same reason that we as humans manage to parse the physical world to the point where we can do science about it.