Posts on: Mathematics
I taught two courses this year that I haven't taught before. One of
them was our 4th-year undergraduate course on mathematical logic,
"Logic, Computability, and Incompleteness". As usual, I ended up writing
my own textbook. Here it
is as PDF and here as
HTML.
Why yet another textbook? Two reasons mainly. One is that many
existing textbooks are addressed at maths students. This shows up not
only in the examples and illustrations, but also in the fact that
comparatively little time is spent motivating, explaining, and
discussing definitions, proof ideas, or results. I wanted more of
that.
A somewhat appealing (albeit, to me, also somewhat obscure) view of
mathematics is the pluralist doctrine that every consistent mathematical
theory is true, insofar as it accurately describes some mathematical
structure. I want to comment on a potential worry for this view,
mentioned in (Clarke-Doane 2020): that
it has implausible consequences for logic.
A famous argument, first proposed in Lucas 1961, supposedly shows that the
human mind has capabilities that go beyond those of any Turing machine.
In its basic form, the argument goes like this.
Let S be the set of mathematical sentences that I accept as true. S
includes the axioms of Peano Arithmetic. Let S+ be the set of sentences
entailed by S. Suppose for reductio that my mind is equivalent to a
Turing machine. Then S is computably enumerable, and S+ is a computably
axiomatizable extension of Peano Arithmetic. So Gödel's First
Incompleteness Theorem applies: there is a true sentence G that is
unprovable in S+. By going through Gödel's reasoning, I can see that G
is true. So G is in S and thereby in S+. Contradiction!
Frege argued that number concept are, in the first place, second-order predicates. When we talk about numbers as objects, we use a logical device of "nominalization" that introduces object-level representations of higher-level properties. In Grundgesetze, he assumed that every first-order predicate can be nominalized: for every first-order predicate F, there is an associated object – the "extension" of F – such that F and G are associated with the same object iff ∀x(Fx ↔︎ Gx). The number N is then identified with the extension of 'having an extension with N elements'. Unfortunately, the assumption that every predicate has an extension turned out to be inconsistent, so the whole approach collapsed.
A common worry about mathematical platonism is how we could know about an independent realm of mathematical facts. The same kind of worry arises for moral realism: if there are irreducible moral facts, how could we have access to them?
Benacerraf (1973) put the problem in terms of causation. Knowledge of maths, he suggested, would require some kind of causal connection between the mathematical facts and our mathematical beliefs, but modern platonists typically don't believe in such a connection.
According to realist structuralism, mathematics is the study of
structures. Structures are understood to be special kinds of complex
properties that can be instantiated by particulars together with
relations between these particulars. For example, the field of complex
numbers is assumed to be instantiated by any suitably large collection
of particulars in combination with four operations that satisfy certain
logical constraints. (The four operations correspond to addition,
subtraction, multiplication, and division.)
In On the Plurality or Worlds, Lewis argues that any account
of what possible worlds are should explain why possible worlds
represent what they represent. I am never quite sure what to make of
this point. On the one hand, I have sympathy for the response that
possible worlds are ways things might be; they are not things
that somehow need to encode or represent how things might be. On the
other hand, I can (dimly) see Lewis's point: if we have in our
ontology an entity called 'the possibility that there are talking
donkeys', surely the entity must have certain features that make it
deserve that name. In other words, there should be an answer to the
question why this particular entity X, rather than that other entity
Y, is the possibility that there are talking donkeys.
Daniel Nolan and I once suggested that talk about sets should be analyzed as talk about possibilia. For simplicity, assume we somehow simply replace quantification over sets by quantification over possible objects in our analysis. This appears to put a strong constraint on modal space: there must be as many possible objects as there are sets.
But does it really? "There are as many possible objects as there are sets." By our analysis, this reduces to, "there are as many possible objects as there are possible objects". Which is no constraint at all!
Some philosophers believe that the second world war is a triple of a thing, a property and a time. Others have argued that my age is a pair of an equivalence class of possible individuals and a total ordering on such classes. It is also often assumed the number 2 is the set {{{}},{}}; that the meaning of "red" is a function from contexts to functions from possible individuals to functions from possible worlds to truth values; that possible worlds are sets of ... sets of properties; and that truth values are the numbers 0 and 1 (aka the sets {} and {{}}).
I once believed that in non-contingent matters, knowledge is true,
justified belief. I guess my reasoning went like this:
How do we come to know, say, metaphysical truths? Not by direct
insight, usually. Nor by simple reflection on meanings, sometimes.
Rather, we evaluate arguments for and against the available
options, and we opt for the least costly position. If that's how
we arrive at a metaphysical belief, the belief is clearly justified
-- we have arguments to back it up. But it may not be knowledge:
it may still be false. Metaphysical arguments are hardly ever
conclusive. But suppose we're lucky and our belief is true. Then it's knowledge: what more could we ask for? Surely not any causal connection to the non-contingent matters.
But now that Antimeta has asked for Gettier cases in mathematics, it seems
to me that there are perfectly clear examples (I've posted a comment over there, but it seems to have gone lost):
Suppose we find a proof, in ZFC, that ZFC is inconsistent. Does
it follow that ZFC is inconsistent?
On the one hand, if we could infer from ZFC
![$m[1]](/texer/images/?format=gif&fontsize=12pt&tex=%5Cmodels)
~Con(ZFC) that ZFC is inconsistent, we could
contrapositively infer the consistency of
ZFC & Con(ZFC) from Con(ZFC); and since ZFC & Con(ZFC) obviously entails Con(ZFC), ZFC & Con(ZFC) would thereby entail its own consistency. Which it only can if it is inconsistent (Gödel's second incompleteness theorem). So it seems that we can only infer that ZFC is inconsistent from the observation that ZFC entails its own inconsistency if we presupposes that ZFC &
Con(ZFC) is inconsistent.
Suppose there are at least proper-class many possibilia. Does it follow that some fusions of possibilia are not members of any set? For the last two years or so I thought it does. My reasoning was that if some of the possibilia correspond one-one with all the sets, then some atoms of possibilia also correspond one-one with all the sets (for there cannot be proper-class many fusions of set-many atoms); but since there are always more fusions of atoms than atoms, it follows that there must be more fusions of atoms of possibilia than sets, and hence that some (in fact, most) of these fusions lack a singleton. This does not take into account atomless possibilia, but I always thought the reasoning would easily carry over, by something like the fact that even with gunk
Brian points to Gabriel Uzquiano's Cardinality Puzzle about Mereology and Set Theory (PDF), which he (Gabriel) introduced a while ago in the now-deceased Philosophy from the 617 weblog. I still don't know enough set theory and mereology to competently discuss the matter, but anyway, it seems to me that perhaps the puzzle can be strengthened, as follows.
I've been invited to this year's German-Italian Colloquium in Analytic Philosophy, for which I've put together some remarks on the philosophy of mathematics: "Emperors, dragons and other
mathematicalia" (PDF). I mainly argue that mathematical sentences should be interpreted as quantifications over possibilia. Technically, this isn't really new. Daniel Nolan in particular has made a very similar suggestion (PDF). What hasn't been emphasized enough, I believe, is that this interpretation not only works from a technical point of view, but is quite attractive for various philosophical reasons. (Unlike Nolan, I argue that it isn't a reform, but a faithful interpretation of mathematics.)
Oh dear.
Returning to philosophy, here is a remark by John Burgess about the possibility of translating ordinary sentences into sentences with seemingly less ontological commitment, as described in Prior's "Egocentric Logic" and Quine's "Variables Explained Away":
Thus whether one speaks of abstract objects or concrete objects, of simple objects or compound objects, or indeed of any objects at all, is optional. Or at least, this is so as regards "surface grammar". My claim is that if children who grew up speaking and arguing in Monist or Nihilist or some Benthemite hybrid between one or the other of these and English, it would be gratuitous to assume that the "depth grammar" of their language would nonetheless be just like that of English, with a full range of nouns and verbs denoting a full range of sorts of objects and connoting a corresponding range of kinds of properties. And any assumption that the divine logos has a grammar more like ours and less like theirs would be equally unfounded, I submit. It is in this sense that I claim any assumption as to whether ultimate metaphysical reality "as it is in itself" contains abstract objects or concrete objects, of simple objects or compound objects, or again any objects at all, would be gratuitous and unfounded. (p.18 of "Being Explained Away" -- Microsoft Word format, use Neevia to convert)
I'm not sure to what extent I agree with that. I do agree that there is something strange about asking whether numbers really exist. Burgess takes this to be the core question dividing nominalism and platonism about numbers. Thus he argues e.g. in "Nominalism Reconsidered" (MS Word again, coauthored with Gideon Rosen) that if nominalists agree that "there are numbers" is true -- while offering a nominalistically acceptable interpretation --, they have actually given up nominalism.
Sometimes I think it's unfortunate that advanced logic and metamathematics usually presuppose various mathematical truths. For instance, in discussions on mathematical realism I've heard people arguing that by the first incompleteness theorem, mathematical truth can't be identified with provability in a formal deductive system. For, those people argue, the first incompleteness theorem proves that for any reasonable formalized system of mathematics, there is a true arithmetical sentence G that is unprovable in the system.
Just for fun, I'm reading Peter Smith's draft of his new book on Gödel's Incompleteness Theorems. So far, it's quite enjoyable. I might say more about it later. But here is something of which I'm not sure it's correct. (I'm also not sure it's false, that's why I'm posting it here.)
Smith shows that not all computable functions are primitive recursive by proving that the antidiagonal of the p.r. functions can't be p.r., even though it is intuitively computable. Having
identified the p.r. functions with functions whose implementation
doesn't require unbounded loops, he then asks why the antidiagonal
function doesn't satisfy that condition:
When I prepared for my exam, I noticed something curious.
Richard Heck, in "The Julius Caesar Objection", claims that
In a letter to Russell, Frege explicitly considers adopting
Hume's Principle as an axiom, remarking only that the 'difficulties here'
are not the same as those plaguing Axiom V [p.274 in Language, Thought
and Logic].
The claim is repeated by Crispin Wright and Bob Hale in the introduction
to The Reason's Proper Study (p.11f., fn.21). The letter Heck,
Wright and Hale refer to is xxxvi/7 from July 1902.
Fictionalism about a certain discourse is the view that statements belonging to this discourse are to be interpreted like statements in fictional discourse.
Now as Brian has observed, on the common account of fictional discourse, "Fictional(Fa)" implies "(Ex)Fictional(Fx)" (even though it normally doesn't imply "(Ex)Fx"). So one might think that on the common account, fictionalism can't do with fewer entities than realism, even though it can do with different entities. However, the common account is not committed to "Fictional(a != b)" implying "(Ex)(Ey)(x != y)". After all, it usually allows for "(Ex)(Fictional(Fx) and Fictional (not-Fx))", so why not allow for "(Ex)(Fictional(x != b) and Fictional(x = b))"? So maybe one could endorse fictionalism about mathematics and the common account of fictional discourse without being committed to an infinity of entities by claiming that all the "numbers" talked about in mathematics are in fact identical.
I vaguely believe that there are no implicit definitions. So I've decided
to write a couple of entries to defend this belief. The defence may well
lead me to give it up, though. Anyway, here is part 1.
Explicit definitions introduce a new expression by stipulating that it be
in some sense synonymous or semantically equivalent to an old expression.
For ordinary purposes this can be done without the use of semantic
vocabulary by stipulations of the form
Today I've been reading Hilbert. I must admit that I don't really
understand his view on the foundations of mathematics. It seems to me that
he always confuses truth with consistency. For example, he writes in his
"New Grounding":
If we can produce [a consistency proof of formalised mathematics], then
we can say that mathematical statements are in fact incontestable and
ultimate truths.
Obviously, Hilbert uses "true" in a very unusual way here: Both ZFC + the
Continuum Hypothesis and ZFC + its negation are consistent. Hence, on
Hilberts account, both CH and its negation are "incontestable and ultimate
truths".
In §27 of Meaning and Necessity, Carnap announces that all
mathematical concepts can be defined without the use of any class
expressions. The basic idea is to use Frege's system, but to replace all
occurrences of class variables with higher order variables. In particular,
the cardinal number of a property F is defined as the second order property
of being equinumerous to F (definition 27-4). "Thus, for example, '2' is a
predicator of second level" (p.117).
Today I found Montague's paper, and it turns out that I was
wrong. Well, Field's presentation was not entirely correct: We
shouldn't take Robinson arithmetic itself as R, but some extension of it
that contains an additional primitive predicate "True" (T, for short). The extension need
not say anything about this predicate. This is why T needn't represent
truth in R. (If R says nothing about T, T either represents nothing at
all or the inconsistent property, depending on how precisely we define
representation.) Montague then shows, very much like Field, that any
theory that contains R -- no matter if it's axiomatizable or not --,
as well as every instance of
Hereby I stipulate that "fb13" is to denote the first human born in the
13th century. Hence it might seems that "fb13 was born in the 13th
century" is analytically true, true by definition. But if analytic truths
are closed under logical implication, "somebody was born in the 13th
century" would also be analytically true. Which it is not.
I don't think tinkering with closure under logical implication will help.
Hereby I stipulate that "fb23" is to denote the first human born in the
23rd century. However, if recent progress in civilization continues, there
might well be no humans in the 23rd century. And if no humans are born in
the 23rd century, "fb23 is a human born in the 23rd century" is false. So
it cannot be true by definition.
Happy new year everybody. I'm still alive, and I still have questions and
comments on the metaphysics of David Lewis. This one is about Lewis'
philosophy of mathematics.
In "Mathematics is Megethology", Lewis argues
for structuralism in set theory: There is no particular relation of
membership, connecting particular things with particular classes. Instead,
there are just two sides of Reality, ordinary individuals on the one side,
proper-class many mereological atoms (called 'singletons') on the other.
Set theory is about all relations on this Reality that satisfy certain
constraints, like 'every individual stands in that relation to a singleton'.