Download READING GUIDE FOR AD INFINITUM This reading selection is

READING GUIDE FOR AD INFINITUM
MD, MATH 111
This reading selection is dealing with some of the epistemological (nature of knowledge) and ontological (status of things) questions one can pose about the mathematically infinite. Is infinity a thing?
Is it a number, or, to ask a slightly different question, are there infinitely big numbers? One frequent
“fix” that mathematicans use to make sense of infinity is to treat infinities as processes that can in
principle (theoretically) be repeated or continued forever– but to understand them, one need only study
what happens finitely far in.
For instance, the sum 1 + 1/2 + 1/4 + 1/8 + ... can be treated as a process for which we always
know how to supply the next term. To understand its sum, we need only look at the finite “partial
sums”: 1, then 1 + 1/2, then 1+ 1/2 + 1/4, and so on. Each of these is a perfectly respectable finite
sum, and the question of the infinite sum amounts to showing that the finite sums never exceed 2,
although they get arbitrarily close. But this is a sort of sleight-of-hand that transfers the burden from
infinity to the “arbitrarily small.” Does it really eliminate the problems with talking about the infinite?
It’s this “and so on,” also encoded as “...” or “ad infinitum” (Latin for “to infinity”), that Rotman
is trying to probe in this passage. Here are some tidbits to help you with the reading.
antinomy (39). An apparent paradox. This term is usually reserved for logical paradoxes.
Zeno’s paradoxes (40). Zeno (−5th century) gave a whole collection of interesting paradoxes (see,
for instance, http://en.wikipedia.org/wiki/Zeno’s paradoxes). Rotman mentions the “arrow in flight”
paradox: suppose you shoot an arrow from here to that target over there. It takes a certain number of
seconds to reach the target. But at any specific instant, it occupies a fixed portion of space (suggested
by the fact that a photograph taken with a fast enough shutter speed will seem to show it standing
still in air). So if at any particular exact time, the arrow is stationary, how can it ever move?
Aristotle, writing about a hundred years later, described the paradox of the arrow this way: If
everything when it occupies an equal space is at rest, and if that which is in locomotion is always
occupying such a space at any moment, the flying arrow is therefore motionless.
I think this paradox is rather subtle and hard to ignore.
potential infinity (41). This is what Stillwell is addressing when he speaks of the Greek hesitancy
to treat infinity as its own thing; to get away from that, an infinite process was simply treated as a
process with “the possibility of its indefinite continuation, and no more—certainly not the possibility
of eventual completion” (Stillwell 51).
constructivists, Kronecker (41). Platonism is, roughly, the philosophical decision to accept the
timeless existence of actual abstract objects, like spheres and squares, that are ideal and perfect and
outside of human activity. (Mathematical) constructivism, on the other hand, only accepts what
can be built up through concrete algorithmic or iterated (repeated) processes. Two other flavors of
mathematical philosophy called intuitionism and finitism are special kinds of constructivism—finitism
insists that constructions only allow finite iteration, and intuitionism is chiefly associated with the
rejection of proofs by contradiction. All of these overlapping mathematical worldviews have in common
a willingness to restrict the kind of mathematical arguments that are allowable; they share a preference
for constructive proofs over abstract “existence” proofs.
Leopold Kronecker is strongly associated with constructivism, and he began to articulate this point
of view largely out of irritation and discomfort with the world of the abstract infinite being pioneered
by Georg Cantor—Cantor being the first to write down a rigorous theory that there were different
sizes of infinity, among other mind-bending ideas. Kronecker wanted to accept the natural numbers
(1, 2, 3, . . .) as a starting point and then only to accept constructions from that foundation.
Gauss repudiation of infinity-as-object (41). Here’s the relevant quote, cited in your reading of
Barrow’s Chapter 2:
Gauss: “I protest against the use of an infinte quantity as an actual entity; this is never allowed in
mathematics. The infinite is only a façon de parler [manner of speaking]...”
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Aristotelian account of motion (41). Roughly speaking, Aristotle denied that time is composed of
individual instants that have no duration but retain a time-like character. That is, the whole is not the
sum of its parts—instants should not be expected to behave as components of a time interval. Motion
is treated as a potentiality for a thing to be somewhere else; when it arrives, that potential has been
realized. Motion needs to be understood as a whole process that can’t be decomposed into isolatable
instants.
infinitesimals (41-2). Calculus provoked quite a philosophical distress call. It was a collection of
techniques that seemed to be founded on tiny little particles of number that were so small that they
didn’t behave like little fractions or behave like zero either. It was only later that Cauchy began to put
this on better footing by coming up with the limit as the variable tends to zero, and Weierstrass finally
nailed the whole matter down with his δ- proofs.
formalism (42). One way out of the vexing problems about what math is all about is to declare that
it doesn’t matter! Just say that you write down axioms, which are to be treated as given, and then you
use agreed-upon rules to make deductions of theorems and so on from your axioms. The celebrated
mathematician David Hilbert is strongly associated with this point of view and is widely believed to
have said something very much like this:
“Mathematics is a game played according to certain simple rules with meaningless marks on paper.”
This quote seems to have been words put in his mouth by his critics, but it’s not that far off of his
stated views.
Cantor’s actually infinite hierarchy of actual infinities (42). Cantor’s work on the infinite found
that there was a distinction between countable and uncountable infinites; but furthermore, he found
that if a set X has a particular cardinality, then its power set (the set of all subsets of X) has strictly
greater cardinality. This means that there is no largest infinity!
dimension (42). The idea of one-dimensional, two-dimensional, and three-dimensional geometric objects is not recent, nor is the debate about what it should mean to go farther up—for instance, an
interesting mathematically based social satire drawing on these questions can be found in the 1884
book Flatland by Edwin A. Abbott. But Cantor’s work on certain oddly-behaved infinite sets ushered
in theories of dimension in greater generality. The main figure here is Felix Hausdorff, who came up
with a rigorous and flexible notion of dimension that’s general enough to treat fractional dimension.
Hausdorff’s dimension theory definitely involves limiting processes in an essential way.
paradoxes and antinomies involving self-reference and infinite aggregates (42). When you
allow just any aggregate (collection of things) to count as a set, this is called “naive set theory” and
it opens the door to various paradoxes. The most famous is Russell’s paradox, from 1901: consider
the set of all sets that do not contain themselves. Does it contain itself or not? Either way, you get a
contradiction.
signifiers (43). Rotman works in a tradition called semiotics, which is a tradition with an ancient
pedigree but these days is largely considered to be a kind of literary theory. In it, one is concerned
with units called signs, which are composed of several parts working together: a signifier, its signified,
and sometimes a referent. To give a simple example, the set of symbols CAT is a signifier, the idea of
a cat is the signified, and an actual cat running around is the referent.
In math, then, the symbol ∞ can be a signifier, as can “. . .,” but the signified is murky, not to
mention the referent.
Platonist orthodoxy applied to Cantor (43). This just means treating the (at one time very
radical) infinite cardinal numbers of Cantor’s work like abstract objects that are “out there” for us to
study.
Kronecker: primality of the integers (45). see constructivism.
a fortiori (45). means “with even stronger reason” or “even more so.”
psychologism, Brouwer, Kant (45-6). Mathematical psychologism is the notion that mathematical laws and objects obey mental and psychological rules. Brouwer’s philosophy of mathematics,
called intuitionism, was about limiting the kinds of allowable arguments until they square with human
intuition—this gives a psychological grounding to mathematical practice. Kant’s philosophy placed the
foundations of mathematics in human intuition of space and time; he argued that this determined the
basic character of mathematics.
READING GUIDE FOR AD INFINITUM
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accessible numbers (46). This is a version of the idea that there isn’t one single thing to be a
number—maybe really big numbers simply have different properties from the familiar ones like seven
and thirteen. Maybe we should consider 444,444,444,469 to have different properties from thirteen
because if we started counting, we could never actually get there—so in some way, it is less “accessible.”
Wittgenstein and surveyable proof (47). Here, Wittgenstein is giving one answer to the question
of what makes one proof better or stronger or more convincing than another. A proof can be called
surveyable if its whole shape and form can be understood—if a kind of visual sense enables someone
reading the proof to understand it as a whole. This is just one criterion—which is partly about
aesthetics, and partly about explanatory value—for what makes a good proof.
Sorites paradoxes (47). These paradoxes serve many purposes, but here the main one is to illustrate
that—never mind the infinite—we even have trouble understanding the difference between Large and
Small. The idea is that if you keep taking one away from a large number, it eventually becomes small,
but when does this happen exactly? It seems reasonable that one less than a large number is still
large, and if this is true then by iteration you eventually arrive at a paradox. Classically, this paradox
is applied to the number of grains of sand in a “heap” or the number of hairs on a head before it is
considered “bald”; Rotman discusses those examples further later in the chapter.
Hilbert (47). see formalism.
metaphysical belief (48). Metaphysics is the branch of philosophy that is concerned with sameness,
identity, and persistence of things. For example, what makes me the same person as the girl who had
my name in New Jersey in 1979? Is it that I still have memories of being that person? If I lost my
memories, would it be less true that I am still the same person?
Or, mathematically, what makes the 30-60-90 triangle and the 45-45-90 triangle two instances of the
same kind of thing (namely, a triangle)? If you’re a Platonist, the answer is simple—triangles are ideal
forms out there, and these are both forms in that class. What Rotman is calling the “inner layer” of
metaphysical assumption here is that triangles never had to be created or invented, because they were
always potential figures, even before there were people to talk about them.
empirically tainted culture (49). Empirical knowledge is that attained through experience. Note
that this contrasts with Platonist knowledge, which is discovered knowledge of abstract objects already
out there, through a human intuition that connects us to ideal forms.
Empirically tainted knowledge, therefore, would be understanding that is colored by encounters with
actual physical objects, rather than perfect ideal objects.
contingencies (49). For one thing to be contingent on another means that it is dependent or relies on
the other. To believe that the history of math is full of contingencies might be, for instance, to believe
that Galois theory would never have been developed if Galois had died at age fifteen instead of twenty.
The usual alternative to contingency is inevitability, as in: even if Galois had never been born,
someone would eventually have invented Galois theory.
psychogenetically, instrumentally (50). Psychogenetic: arising as part of a psychological process.
Instrumental: being a means to an end.
ideogram (51). A symbol—an icon rather than a word—that represents an idea.
intersubjective, agency (52). A subject or an agent is the entity that does, experiences, or performs
something. An intersubjective activity occurs between two or more conscious agents. Agency is the
property or condition of being able to act. So it is a reasonable question to ask about counting: who
or what has the agency? If counting is imagined as an action (enumerating objects), then one might
well ask if counting makes sense without someone to do the work of reciting or ticking off numbers.
formal arithmetic (52). This term refers to placing arithmetic inside a formal system: certain axioms
and rules of deduction which define numbers and in which one can prove theorems like “15+179 = 194.”
declare by fiat (53). If something is true by fiat, then it is true simply because some authority
declared it to be so. A fiat is an official order.
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mathematically acceptable process of idealization (59). Idealizing, or abstracting, something
means taking its salient (defining or most relevant) features and discarding its incidental or surface
features. It means distilling the essence of a thing—often an actual thing in the world—and creating a
notion that is outside of experience.
For instance, if a grocery store worker is asked to stack oranges as efficiently as possible, this can be
turned into a mathematical problem by treating the oranges as perfect spheres. For another example,
there is a mathematical subject called “billiards” in which the tables are planar polygons, the ball is
a point mass that moves without friction and with perfect reflection in the walls, and there are no
pockets—clearly, an “idealized” game of pool.
In order to turn a question into something approachable by mathematics, it is typically necessary to
make a move or series of moves of this kind.
Finitism as discursive ascription, posterior and privative move (59). Discursive means relating
to discourse, which is conversation or discussion or intersubjective rhetoric. To ascribe means to assign.
So “finitism as discursive ascription” refers to certifying objects or processes as acceptably finite, or
free of the taint of the mysterious infinite.
Posterior means at the rear, or after-the-fact. Privative means about negation. So calling the finite
a “privative and posterior move” means that it is defined as not infinite, and that it only makes sense
after the infinite is already understood.
infinitism inuring itself against any criticism of its status (59). To inure is to habituate or
accustom, often to inoculate or protect oneself against something. You can become inured to something
unpleasant, so that at least you are used to it and it no longer seems so bad. In this case, we can become
inured to the idea of infinity—so accustomed to it as a background concept that we no longer question
it.
If the finite is defined as being not-infinite, then the infinite is taken for granted as being already
understood. If we accept that “God made the natural numbers,” then we have no leg to stand on if we
seek to question infinity.
a priori and synthetic (60). Immanuel Kant (1724-1804) was an enormously influential philosopher
who wrote about a huge range of topics, including “pure reason,” which dealt extensively with the
philosophy of math. Two important distinctions for him are analytic (true in a self-contained way) vs.
synthetic (true or not true due to causes exterior to the terms in the statement), and a priori (before
the fact) vs. a posteriori (after the fact).
Example of analytic statement: all bachelors are unmarried. (This is a tautology—it’s true because
of what “bachelor” means.)
Example of synthetic statement: all men have beards. (Having a beard is not part of what it means
to be a man.)
Example of a priori statement: if all men are mortal, and Socrates is a man, then Socrates is mortal.
(This is seen to be true by deduction, not by checking any properties of Socrates in person—it is
necessarily true.)
Example of a posteriori statement: pink elephants exist. (You can verify this by experience, and
otherwise you might not believe it.)
Kant’s much-cited, and provocative, claim is that mathematical statements like 7 + 9 = 16 are
synthetic a priori: they are not true by definition, but they are nonetheless true without being subject
to testing or empirical verification.
to violate the relevant integrity of the object (61). Here the best example I can think of is the
following joke: a pure mathematician is one of a long line of experts hired by UC Davis to help increase
the milk production of cows. Several aggie specialists have come up with their best suggestions, but
the mathematician announces a solution with a far higher yield of milk, so high that the other experts
are stunned. They all turn to the mathematician expectantly, waiting to hear how to get that much
milk out of the local livestock. “Easy!,” declares the mathematician, “First assume a spherical cow...”
In other words, sometimes abstractions that are helpful for making mathematically approachable
problems lose the salient features of the objects they are trying to describe, and so become useless.