Transcript Intertheoretic Reduction and Explanation in Mathematics

Intertheoretic Reduction and
Explanation in Mathematics
BILL D’ALESSANDRO
UNIVERSITY OF ILLINOIS -CHICAGO
Outline
1.
What is intertheoretic reduction, and why care about it (in math)?
2.
A (somewhat) Nagelian approach to reduction
3.
Reduction and explanation
◦
An explanatory reduction: The classical theory of equations and Galois theory
◦
◦
4.
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:
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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 𝑇1 is reducible to 𝑇2 , then 𝑇2 explains, and hence improves our understanding of,
some of the phenomena described by 𝑇1 . 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 𝑇1 being reducible to 𝑇2 and we have a view about what this involves, this might place
some interesting constraints on what kinds of things 𝑇1 and 𝑇2 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:
1.
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.)
2.
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:
1.
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 𝑇1 is reducible to 𝑇2 , then the objects and properties in
the domain of 𝑇1 must be identical to/composed of/constituted by/supervenient on the objects and
properties in the domain of 𝑇2 .
2. 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 ‘ 𝑇 interprets 𝑆’ means that the language of 𝑆 is translatable into the language of 𝑇 in such a way
that 𝑇 proves the translation of every axiom of 𝑆” (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 “𝑇1 is reducible to 𝑇2 ” means basically that
(an appropriately formalized version of) 𝑇2 interprets (an appropriately formalized version of) 𝑇1 .
◦ 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 𝑛th roots.
◦ The single root of a linear equation 𝑎𝑥 + 𝑏 = 0 is given by 𝑥 = − 𝑏 𝑎 . (Trivial!)
◦ The two roots of a quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 are given by the “quadratic formula”,
−𝑏 ± 𝑏2 − 4𝑎𝑐
𝑥=
.
2𝑎
(Less trivial, but known since antiquity.)
◦ Finding solution formulas for equations of degree ≥ 3 turns out to be much harder. The degree 3 and
4 cases were solved by Italian mathematicians in the 16th 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 ≥ 5 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 5 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 𝑥 2 + 𝑦 2 − 1 is the set of points 𝑥, 𝑦 such that 𝑥 2 + 𝑦 2 = 1—
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:
“ 𝑥 , 𝑥, 𝑦 is a single set that codes the identities of the two objects 𝑥 and 𝑦, 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 2 ∈ 3 ).
 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 𝐴 and 𝐵 are sets, then there is a oneto-one correspondence between the cartesian products 𝐴 × 𝐵 and 𝐵 × 𝐴. The central idea in
the proof of this fact is the old observation that a rectangle of 𝑛 rows of 𝑚 dots contains 𝑛 ∙ 𝑚
dots, but turned on its side it contains 𝑚 ∙ 𝑛 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:
1.
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?
2.
◦
Seemingly not the existence or nonexistence of explanatory proofs.
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Something “metaphysical” about the relationship between the two domains of mathematical objects? (Cf. Pincock 2015)
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Or something else (e.g., epistemological/logical)?