Intertheoretic Reduction and Explanation in Mathematics - PowerPoint Presentation

Intertheoretic Reduction and Explanation in Mathematics Bill D’Alessandro University of Illinois-Chicago

Outline What is intertheoretic reduction, and why care about it (in math)? A (somewhat) Nagelian approach to reduction Reduction and explanation An explanatory reduction: The classical theory of equations and Galois theory Possible bonus: Classical algebraic geometry and scheme theory A non-explanatory reduction: Number theory and set theory Further questions

The concept of intertheoretic reduction Intertheoretic reduction : Roughly, a form of theory succession in which the successor (“reducing”) theory is substantially similar to or continuous with the predecessor (“reduced”) theory, e.g. at the level of laws, structure or ontology. We often get reductions when the predecessor theory is partially or approximately correct , or when it’s true only within a relatively limited domain, for instance. Some familiar (though sometimes controversial) examples from the sciences: Classical thermodynamics and statistical mechanics Kepler’s theory of planetary motion and Newtonian mechanics Newtonian mechanics and Einsteinian relativity Chemistry and quantum mechanics Mendelian genetics and biochemistry The serious study of intertheoretic reduction starts with Ernest Nagel’s The Structure of Science (1961). Since then, the concept of reduction has been a mainstay of Anglophone philosophy of science.

Why care about reduction? Reduction has proved to be a useful concept in philosophy of science in a few ways. For instance: Reduction is one of several important forms of theory succession. (It contrasts with replacement , which is more radical, and also with relatively gradual and conservative kinds of theory change.) So it plays a big role in the project of classifying succession relations . Reduction is closely related to issues of scientific explanation and understanding. Typically—or maybe even necessarily—if is reducible to , then explains, and hence improves our understanding of, some of the phenomena described by . So the study of reduction promises to shed light on these issues.Understanding reduction may tell us something about the nature of scientific theories. (If we’re committed to being reducible to and we have a view about what this involves, this might place some interesting constraints on what kinds of things and can be and how they’re related to one another—e.g. metaphysically, epistemologically, or logically.)The notion of reduction can help illuminate the history and practice of science, by showing why the scientific community (or an individual scientist) regards a theory in a certain way at a certain time.  

Why care about reduction in math? Almost everyone acknowledges the existence of reducibility relations in mathematics. For instance, we’ve all heard plenty about “set-theoretic reduction”. But philosophers of math have had much less to say about various important aspects of reduction than their counterparts in philosophy of science. This is unfortunate! All the reasons to care about reduction in science are equally good reasons to care about reduction in math. Viz., Because it’s worthwhile to classify different kinds of succession relations between mathematical theories. Because mathematical explanation and understanding are extremely important.Because we’d like to better understand the nature of mathematical theories and the relations between them.Because we want useful tools for analyzing the history and practice of mathematics. My dissertation project is an attempt to think seriously about some of these issues.

Rantala on reduction and explanation Veikko Rantala : [I]n the philosophy of science the notions of explanation and reduction have been extensively discussed… but there exist few successful and exact applications of the notions to actual theories, and, furthermore, any two philosophers of science seem to think differently about the question of how the notions should be reconstructed. On the other hand, philosophers of mathematics and mathematicians have been successful in defining and applying various exact notions of reduction (or interpretation), but they have not seriously studied the questions of explanation and understanding. (1992, 47)

Goals of the talk The two main things I’ll be doing here: Asking how we should conceptualize reductions in math, and suggesting that an approach similar to Nagel’s is preferable. (On Nagel’s view, reduction is essentially a linguistic and logical relation, not a metaphysical one.) Arguing for a novel distinction—which seems to have no parallel in empirical science—between mathematical reductions that are substantially explanatory and those that aren’t. I illustrate the distinction with an example of each kind. (The reduction of the classical theory of equations to Galois theory is explanatory, while the reduction of number theory to set theory isn’t.)

How to think about reduction A couple assumptions I’ll be making about the nature of reduction: As Nagel held, reduction is essentially a linguistic and logical relationship between theories, not a metaphysical relationship between families of things. This runs counter to views according to which, if is reducible to , then the objects and properties in the domain of must be identical to/composed of/constituted by/supervenient on the objects and properties in the domain of .Reduction doesn’t essentially involve explanation. That is, there can be (and I think there actually are) reductions in which the reducing theory doesn’t explain the phenomena described by the reduced theory.This runs counter to Nagel’s own view, according to which “Reduction... is the explanation of a theory or a set of experimental laws established in one area of inquiry, by a theory usually though not invariably formulated for some other domain” (1961, 338). 

How to think about reduction An approach that meets these criteria, and which several authors have found attractive on independent grounds, is to identify reduction with the model-theoretic notion of (relative) interpretability . “Intuitively ‘ interprets ’ means that the language of is translatable into the language of in such a way that proves the translation of every axiom of ” (Berarducci 1990, 1059). This seems to get at the core idea of reduction without bringing along unwanted metaphysical or epistemological baggage. So I’ll assume from now on that “ is reducible to ” means basically that (an appropriately formalized version of) interprets (an appropriately formalized version of) .Informally, I’ll just try to show that there’s some reasonable correspondence between the vocabulary of the two theories that’s suggestive of interpretability. 

And now, On to the examples.

Galois theory and the classical theory of equations The theory of equations : a collection of classical problems and results about polynomial equations—traditionally, polynomial equations in a single variable with integer coefficients. The oldest and most famous of these problems involves finding the set of solutions to a given equation “in radicals”, that is, as a function of the equation’s coefficients involving only basic arithmetical operations and th roots. The single root of a linear equation is given by (Trivial!)The two roots of a quadratic equation are given by the “quadratic formula”, (Less trivial, but known since antiquity.) Finding solution formulas for equations of degree turns out to be much harder. The degree 3 and 4 cases were solved by Italian mathematicians in the 16 th century, through a painstaking process of trial and error involving various substitutions and manipulations.  

Galois theory and the classical theory of equations After these successes, it was hoped that solution formulas for higher-degree equations would soon follow. But nobody was able to find them! Eventually it was conjectured that equations of degree simply weren’t generally solvable in radicals, though it was far from clear at first why this should be the case. Lagrange took a major step forward in 1770, realizing the importance of studying permutations of the roots of equations. Building on Lagrange’s approach, Abel and Ruffini proved the unsolvability of the general quintic in the early 19th century. 

Galois theory and the classical theory of equations Finally, Galois managed to put the whole question to rest through the use of a new set of theoretical tools. In this approach, now known as Galois theory, one studies a certain algebraic object associated with a given polynomial (namely its “Galois group”, the group of permutations of the roots). If the Galois group of a polynomial has a certain property (also known as ‘solvability’), then the polynomial is solvable in radicals; if not, then no solution formula exists. As it turns out, polynomials of degree less than five always have solvable Galois groups, but this is false for higher-degree polynomials. Hence Galois was able to show that equations of high degree are in general not solvable in radicals.

Galois theory and the classical theory of equations This case looks like a good candidate for reduction! Informally , a theory is reducible to its successor when the successor extends the original while preserving some of its essential concepts and results. And that’s what we have here. Galois theory retains central elements of the classical theory of equations—e.g. the concepts polynomial equation and solvable in radicals, and the theorem that equations of degree at most four are always solvable—while greatly expanding and modifying the earlier theory’s repertoire of ideas, methods and results. We also have theoremhood-preserving translations of the sort we expect. In light of the relationship between the classical theory of equations and Galois theory, we can successfully translate claims about the existence of solution formulas into claims about the solvability of Galois groups.

Galois theory and the classical theory of equations Importantly, this is also an explanatory reduction. Galois theory doesn’t just give us some new knowledge; it also provides a deeper understanding of the solvability of polynomial equations (and related phenomena): “ Failure to solve the quintic led to Lagrange’s theory of equations of 1770, which emphasized permutations of the roots and implicitly contained some ideas of group theory. This was followed by attempts to prove unsolvability of the quintic, by Ruffini and Abel, which led to further understanding of permutations and hinted at the theory of fields. Finally, in 1830, a complete understanding of solvability (of equations) was achieved when Galois brought to light the underlying concept of solvability of groups.” (Mikhalev & Pils 2013, 540)“Not only does [Galois theory] prove that the general quintic has no radical solutions, it also explains why the general quadratic, cubic and quartic do have radical solutions and tells us roughly what they look like“ (Stewart 2007, 116)“By the early 19th century no general solution of a general polynomial equation ‘by radicals’... was found despite considerable effort by many outstanding mathematicians. Eventually, the work of Abel and Galois led to a satisfactory framework for fully understanding this problem and the realization that the general polynomial equation of degree at least could not always be solved by radicals.” (Baker 2013, 3) 

Classical algebraic geometry and scheme theory Classical algebraic geometry is “the study of geometry using polynomials and the investigation of polynomials using geometry” ( Kollár 2008, 363). Its primary objects of study are algebraic varieties = zero sets of systems of polynomial equations. These zero sets have a natural interpretation as geometric objects.For instance, the zero locus of the polynomial is the set of points such that —that is, the set of points comprising the unit circle. So the unit circle is an example of an algebraic variety. Classical algebraic geometry starts with Descartes and Fermat in the 17th century, with the realization that polynomial algebra can be used to solve interesting problems in geometry. It continued to develop and amass results for the next several hundred years. In the 1960s, Alexander Grothendieck put the subject on a new foundation by introducing schemes, more abstract kinds of objects in terms of which classical varieties can be defined.“[Scheme theory] is the basis for a grand unification of number theory and algebraic geometry, dreamt of by number theorists and geometers for over a century. It has strengthened classical algebraic geometry by allowing flexible geometric arguments about infinitesimals and limits in a way that the classic theory could not handle. In both these ways it has made possible astonishing solutions of many concrete problems” ( Eisenbud & Harris 2000, 1).  

Set theory and number theory Since it’s widely agreed that number theory (along with most of the rest of modern mathematics) is reducible to set theory, I’ll take this point for granted. But is this an explanatory reduction? Does the set-theoretic viewpoint show us why the facts of arithmetic are true, or improve our understanding of numbers? I’ll argue that it doesn’t. I think this is a commonsensical idea. Other authors have pointed out that set-theoretic reductions are often conventional or artificial, and hence that we shouldn’t expect them to be a source of new insights about the subject matter.E.g., Michael Potter on the ordered pair: “ is a single set that codes the identities of the two objects and , and it is for that purpose that we use it; as long as we do not confuse it with the genuine ordered pair (if such there is), no harm is done. In other words, the ordered pair as it is used here is to be thought of only as a technical tool to be used within the theory of sets and not as genuinely explanatory of whatever prior concept of ordered pair we may have had” (2004, 65). 

Set theory and number theory Something similar seems to be true for set theory and number theory. For one, the set-theoretic viewpoint doesn’t give us any new arithmetical knowledge (by design). Set theory is conservative over arithmetic—what we can prove in “set-theoretic number theory” is exactly what we can prove in ordinary number theory (aside from “junk theorems” like Also, set theory obviously doesn’t give us a more convenient system of representations that’s advantageous for, e.g., calculation or problem-solving. Even writing out simple statements or proofs in set-theoretic language is horribly unwieldy. Since set theory doesn’t tell us anything new about numbers, and since it doesn’t make the things we already knew any easier to see or more convenient to work with, this seems to be a non-explanatory reduction.  

Objections But not everyone agrees. “ Reducing arithmetic to set theory has explanatory, as well as ontological, value. For, in the light of the reduction, our understanding is advanced through exhibition of the kinship between theorems of arithmetic and theorems in other developments of set theory (in particular, branches of abstract algebra ).” ( Kitcher 1978, 123)An interesting claim, but also obscure. Without knowing what Kitcher has in mind here (he doesn’t elaborate), it’s hard to know whether he’s right!

Objections “Now suppose you take a set theoretic perspective and again ask why multiplication is commutative. Here an answer is forthcoming: because if and are sets, then there is a one-to-one correspondence between the cartesian products and . The central idea in the proof of this fact is the old observation that a rectangle of rows of dots contains dots, but turned on its side it contains dots. I take it that this explains why multiplication is commutative” (Maddy 1981, 499).This seems problematic for a couple reasons.First, Maddy’s explanation sneaks in geometry—”rectangle” is not a concept of pure set theory!Second, this seems to get the order of determination backward. The set-theoretic definitions of the arithmetical operations were chosen in order to validate our pre-existing arithmetic knowledge. It seems odd to suggest that such a deliberate, conventional choice of definition would explain the very facts it was meant to reproduce. 

Further questions If there are both explanatory and non-explanatory reductions in math, then some interesting questions arise: Why do non-explanatory reductions occur in math, but apparently not in the sciences? (Or do they?) What makes a given reduction either explanatory or not? Seemingly not the existence or nonexistence of explanatory proofs.Something “metaphysical” about the relationship between the two domains of mathematical objects? (Cf. Pincock 2015)Or something else (e.g., epistemological/logical)?