Download Session 12: Intuitionism

Philosophy of Mathematics
Ways Forward from Logicism & Intuitionism
• Usually we just use numbers as if they were all given to us in a
lump. But it helps to sort them out and be clear about which
ones we need for different jobs.
• The natural numbers, ℕ, are the basic counting numbers.
• This is fine if you just want to add and multiply.
• The integers, ℤ, are the negative and positive whole numbers,
plus zero.
• This is handy if you want to be able to subtract.
• The rationals, ℚ, are the positive and negative fractions, plus
zero.
• Here we can also divide.
• The reals, ℝ, are the numbers you need to identify the points
in a continuous line. Question: are the reals and the rationals
the same numbers?
Know Your Number System!
• Extensionality: Two sets that have exactly the same elements
are the same set.
• Regularity: All sets are well-founded.
• Replacement: This is an axiom schema that allows us to “build
up sets from other sets” by saying things like “X is the set of
all x such that…” without falling into Russell’s Paradox.
• You’ll sometimes see an “Axiom of Separation” as an
alternative.
• Null set: there exists a set of which nothing is an element
• Unordered pairs: given sets x and y, there exists a set {x, y}
• Union: given sets x and y, x union y exists.
• Power set: given x, its power set exists.
• Infinity: A specific infinite set exists.
A Version of the ZF Axioms
The Principle of Comprehension licenses us to declare:
• “S is the set of all sets that don’t contain themselves”
• Is R a member of itself?
• If it is, then it must satisfy the condition of not being a member of
itself and so it is not.
• If it is not, then it must not satisfy the condition of not being a
member of itself, and so it must be a member of itself.
• Since by classical logic one case or the other must hold –
either R is a member of itself or it is not – it follows that the
theory implies a contradiction.
• S is a contradictory object; any theory that allows it is absurd: ex
falso quodlibet.
Russell’s Paradox
PART 5
LOGICISM IN RUSSELL &
WHITEHEAD’S PRINCIPIA
MATHEMATICA
• Reduction of mathematics to a founding discipline, logic
& set theory
• Requirement: founding discipline is consistent
• Arithmetic would be branch of logic
• Logic - picks out generality
• no special domain of knowledge - but all domains
• objects of any kind can be numbered
• all objects belong to a class
• Numbers themselves need not be objects
Logicism
Principia Mathematica Russell & Whitehead (1910-13)
• Functions and statements can be arranged into hierarchy
• Sets, or ‘classes’, at different level (Type 1) from individuals (Type 0)
• Sets of sets (Type 2 ) at different levels from sets (Type 1)
• etc.
• Avoids the Paradox
• No propositional function can be defined prior to specifying the function's
scope of application
• Domain of objects, individuals, must first be specified before function is
defined
• Avoids problem of ‘illegitimate totalities’
• “Whatever involves all of a collection must not be one of the collection.”
• Numbers become classes of classes rather than objects
Theory of Types
• This axiom guarantees the existence of a set that can be
interpreted as “the set of natural numbers”.
• There are various ways to build sets that can be interpreted as
natural numbers – the details aren’t important.
• This axiom plays a similar role to induction in Peano Arithmetic.
• Without it we have finite set theory, which would appear to be a
theory of the potential infinite only.
• It can say “this even number is the successor of an odd number”
for any even number you choose.
• It can say “in this set of even numbers, every one is the successor
of an odd number” for any set of numbers you like, however large.
• But it can’t say “every even number is the successor of an odd
number”.
• The point to note is that we can’t get our first infinite set just
by juggling with finite sets. We must “import” or “assert” it.
The Axiom of Infinity
Stephen Shapiro:
"The contrast with Frege is stark. Frege proved that each natural number
exists, but his proof is impredicative, violating the type restrictions.
Russell had to assume the existence of enough individuals for each
natural number to exist. This puts a damper on logicism."
Discuss: Logicism failed because it couldn’t guarantee its objects
existed
But ... Gödel’s incompleteness theorems also struck a blow
1. Always at least one sentence that is true but unprovable
2. No formal system can prove its own consistency
Gödel: Principia Mathematica ‘a considerable step backwards as
compared with Frege’ (1944)
Failure of Logicism?
‘[Russell & Whitehead’s] attempt ran against the difficulty
that would have supplied the only valid ground for
Frege's insistence that numbers are genuine objects, the
impotence of logic (at least as they understood it) to
guarantee that there are sufficiently many surrogate
objects for the purposes of mathematics, forcing them to
make assumptions far from being logically true, and
probably not true at all:
to secure the infinity of the natural-number sequence,
they had to assume their axiom of infinity, ... .'
Dummett on PM
Most philosophy of mathematics is post-Fregean, largely
accepting the semantics but concerned over the status of
mathematical objects (e.g. fictionalism)
Dummett: 'we may conclude that Frege proved all arithmetical
propositions that do not require the existence of infinitely
many natural numbers are analytic’
(Frege returned to Kantian ideas)
Other, more radical options:
1. Gödel - special arithmetical intuition
2. Empiricism or Naturalism – dispense with the special status
of arithmetic & mathematical objects
Quine – maths is subordinate to physics
3. Intuitionism – classical logic & mathematics is a problem
Ways forward
Intuitionism
• Dutch mathematician – early 20th Century
• Involved in controversies over foundational programme with
David Hilbert
• Mathematics is based on the mental construction of
mathematical objects
• Subjective idealism?
• Brouwer: Kant is 'old form of intuitionism'
• Maths is not founded on logic
• The fundamental faculty for arithmetic is intuition
• Succession in time & counting
• But broader faculty of grasping first principles – not just
spatiotemporal
LEJ Brouwer (1881-1966)
• Constructivism
• Mathematical objects are constructed by (human) activity
• Entities are equivalent to their process of construction (steps)
• Intuitionism
• Series of mathematical positions regarding real numbers & nature of
continuum
• Alternative Logic based on semantic consequences of proof, not truth
• Not every mathematical statement has determinate truth-value
• Vs double negation elimination & law of excluded middle
• Reformulation of classical mathematics sought
• New Interpretation of real numbers & continuum
• Infinity as potential infinity
• to grasp an infinite structure is to grasp the process that generates it
• NOT A COMPLETED TOTALITY
Intuitionism & Constructivism
Classical logic is fine for empirical statements – number of stars or
comets is not generated by counting
"A realistic conception of the external world assures us that, once we
are satisfied that the concept is sharp in these respects, we need do no
more to guarantee determinate truth-values for quantified statements
involving it, statements to the effect that there is a comet satisfying
some condition, or that all comets satisfy some other condition.
In general, the determination of the truth-values of our sentences is
effected jointly by our attaching particular senses to them and by the
way things are. We do not need to specify what comets there are,
once we have rendered our concept of a comet sharp: reality does that
job for us, and reality therefore determines the truth or falsity of our
quantified statements. So, at least, realism assures us.
Dummett on intuitionism 1
Classical logic is inadequate for mathematical statements, where numbers
are generated:
"Hardly anyone is realist enough about mathematics to think in the same
way about quantified mathematical statements.
A fundamental mathematical concept, say real number, which
determines the domain of quantification of a mathematical theory, must
indeed have a criterion of application and a criterion of identity. Given a
mathematical object, specified in some legitimate way, we must know
what has to hold good of it for it to be a real number; and, given two
such specifications, we must know the condition for them to pick out the
same real number.
Few suppose, however, that, once these two criteria have been fixed,
statements involving quantification over real numbers have thereby all
been rendered true or false; to achieve that, it would be generally agreed
that further specifications on our part were required, in some fashion
circumscribing the totality of real numbers and laying down what real
numbers there are to be taken to be." [197]
Dummett on intuitionism 2
• Disjunction – same form of expression but I has a determinate truthvalue while II does not
• Classical – II is true or false, we just don’t know the case yet
• Intuitionistic – II is not well-defined because we cannot construct
both sides of the disjunction
• "There is an infinite number of prime pairs." (prime pair <11,13>)
• Either true or false? Neither true nor false?
• "If a Well formed formula is not constructively provable using the
rules of inference, then it is not meaningful, for we cannot know what
its truth consists in, since we have no justification."
Motivation for intuitionism
• Classical – says ‘A’ andmeans ‘A is true’
• Intuitionist – says ‘A’ but means ‘we have a proof for A’
• Classical:
‘Not all x’s are F’ therefore ‘there is at least one x that is not F’ – VALID
• Intuitionistic:
‘We do not have a proof that all x’s are F’ therefore ‘we have a proof that
there is at least one x that is not F’ INVALID
Quantification
differences
• double negation introduction is allowed
A Ⱶ ¬¬A
• If we know that A, then it follows that we know that it is
not the case that we do not know that A
• double negation elimination is not allowed
¬¬A Ⱶ A
• If we do not know that we do not know A - we cannot
conclude A
Double negation
Classical vs Intuitionist
• Infinite decimal expansion of real numbers (or Cauchy
sequences of rationals or continued fractions)
• Cannot assume these numbers exist such that nth digit is fixed
• Early Brouwer: Only a real number constructed according to a
rule or law is legitimate
• Late Brouwer: Free choice sequences by creative subject
• Dummett:
"As soon as we consider a domain which cannot be effectively
enumerated, the quantifiers need to be explained ... by
reference to a proof that a given object belongs to the domain."
• Must intuitionism be wrong if it demands revisions to
mathematical practice?
Real numbers
Continued Fractions
Hilbert wrote these words in 1922, responding to the
intuitionist programme of Weyl, Brouwer and others.
His aim, then, is to preserve the achievements of the C19
against skepticism and paradox.
1. Express statements about infinite objects as finite
strings of symbols in a precisely-defined formal
language.
2. Express rules of inference as precisely-defined
operations on symbol strings.
3. Derive theorems from the axiom strings by finitely
many applications of rules of inference.
4. Prove that the rules of inference produce no
contradictory sentence, by purely finite reasoning about
strings of symbols.
Hilbert’s Programme
APPENDIX ON LOGIC
• What we’ve just described is syntax: mere symbol-shuffling.
• We like to think there’s another layer: the meaning of the
symbols. This is their semantics.
• Without this, a formal language is just a kind of abstract game. To
matter to us, it must be about something.
• Still, we can use it to state (purely abstract) axioms and proofs of
theorems.
• Given a set of formally-encoded axioms, the set of all the
theorems we can prove from them is called their theory.
• This is captured more formally by the idea of an interpretation
of the pure symbolic syntax.
• We find a model – something outside the theory that it’s
suppposed to be “about”.
• Then we map the symbols of the language, one by one, onto
features of the model.
Syntax vs Semantics
• A theory is consistent if it doesn’t prove any
contradictions.
• If you can produce a proof of “P” and of “not P”, the theory
is inconsistent.
• Assumption: if a theory is inconsistent, it has no model.
• That is, there are no “real” contradictions.
• It follows that if a theory demonstrably has a model, it’s
consistent.
• Ex falso quodlibet
• It’s a feature of classical logic that if you can prove a
contradiction, you can use that contradiction to prove any
other statement you like, even if the two have nothing to do
with each other.
Contradiction
• Suppose we, who are working in a formalized language, have a
theory that we interpret as being about sea creatures.
• Suppose this theory can prove “Whales are fish” and also, by a
different line of argument, “Whales are not fish”.
• We may now prove “Humans are fish” as follows:
• “Whales are fish” is true, so “Either whales are fish or humans are
fish” is also true.
• This is because “either X or Y” is true if X is true, regardless of
whether Y is true.
• But “Whales are not fish”
• Well, “Either whales are fish or humans are fish” and “Whales are
not fish”. So it must be that humans are fish!
• This is because when “either X or Y” is true and X is false, Y must
be true.
• Nobody thinks this is a good proof that humans are fish. But
it’s not so easy to point to one step that causes the trouble.
• This formal language uses letters to represent propositions, such as
“Whales are fish”.
• The built-in symbols, besides letters, are ¬ (“not”), | (“or”) and &
(“and”), along with parentheses “(“ and “)”.
• The rules for forming expressions are:
•
•
•
•
•
Every letter on its own is an expression
If X is an expression, so is ¬(X)
If X and Y are expressions, so is (X | Y)
If X and Y are expressions, so is (X & Y)
Nothing else is an expression
• We then need a set of rules for combining expressions into proofs. These
are usually pretty short but in the interests of space we won’t set this out
here.
• Here is the proof we just did on the previous slide:
P
(P | Q)
¬(P)
(¬(P) & (P | Q))
Q
Propositional Logic
• Predicate logic allows us to “break apart” propositions a bit.
It’s a bit more complicated as a result.
• To express “Whales are fish”, we might write
∀(𝑥)(𝑤 𝑥 → 𝑓 𝑥 )
• Which we would read aloud as “for every creature, x, if x is a
whale (“w(x)”) then x is a fish (“f(x)”).
• Predicate logic gives us a way to express mathematical
statements. For example, let’s change the model:
• Let the variable, x, range over the natural numbers instead of
animals.
• Let w(x) mean “x is an even number greater than 2”
• Let f(x) mean “x is the sum of two prime numbers”
• Now our symbols express Goldbach’s Conjecture!
Predicate Logic