philosophies of mathematics -...

Philosophies of Mathematics

The Search for Foundations

Foundations

• What are the bedrock, absolutely certain, immutable truths upon which mathematics can be built?

• At one time, it was Euclidean Geometry.

Disaster, and Attempts at Recovery

• Non‐Euclidean Geometry• Examples of space‐filling curves and continuous, nowhere‐differentiable functions brought into question our geometric intuitions.

• Solutions:– Base real numbers on positive integers– Base positive integers on sets.

Disaster, and Attempts at Recovery• Peano Axioms:

– 1. Zero is a number. – 2. If a is a number, the successor of a is a number. – 3. Zero is not the successor of a number. – 4. Two numbers of which the successors are equal are themselves equal.

– 5. (Induction axiom.) If a set X of numbers contains zero and also the successor of every number in X, then every number is in X.

• From these you can build the Whole Numbers, or prove that some object you create behaves like the whole numbers.


• From the Whole Numbers, you can build:– Integers as equivalence classes of ordered pairs of whole numbers: iff

– Rational numbers as equivalence classes of ordered pairs of integers (whose second elements are never 0): iff

– Reals as carefully defined infinite sets of rational numbers (such as Dedekind cuts or Cauchy sequences).

– And so on to complex numbers, quaternions, etc.


• From set theory, you can build the counting numbers:

and so on. . . .


• Actually, to do this you need the Axiom of Infinity in your Axioms for Set Theory:

• and

• This gives you an infinite set that is also “inductive” – you can define and prove things by induction.


• Anyway, after the discovery of non‐Euclidean geometry and “monsterous” functions in calculus, the foundations of mathematics – the bedrock, immutable truth – moved from geometry to arithmetic and eventually to set theory.

• If we could just prove set theory, which was very close to being logic, was a good foundation, we’d be back to having a strong foundation for all of mathematics.


• Gottlob Fregeattempted to show that arithmetic (i.e. the natural numbers) could be built from (informal) set theory.

• Informal Set Theory assumed you can build any set you want.


• Bertrand Russell discovered “Russell’s Paradox”.

• Suppose A is the set of all sets that do not contain themselves as elements.

• Does • This showed that you

can’t just go creating any old sets you want.


• Russell’s paradox, and other similar problems that came to light in “informal” set theory, caused the Crisis in Foundations. There were three responses from within mathematics, which have come to dominate historical accounts of the philosophy of mathematics.– Logicism (Russell and Whitehead)– Constructivism or Intuitionism (Brouwer)– Formalism (Hilbert)

Logicism

• Attempted to reduce mathematics to logic. • Principia Mathematica , by Russell and Whitehead.

• Two Problems:– By the time they were done, the foundational logic was a holy mess – you couldn’t really claim it was just “rules of correct reasoning.”

– The same thing that got Hilbert eventually also applied to this work. Stay tuned.

Formalization

Page 379

Two Very Patient Men

Constructivism or Intuitionism

• Founded by LuitzenEgbertus Jan Brouwer.

• Sort of.• Also Kronecker, Poincaré, Weyl.

• Elaborated on by Brouwer’s student, Heyting.


1. Took the counting numbers as foundational and intuitive.

2. Felt mathematics was about what human minds can construct, and not about language.

3. Rejected the Law of the Excluded Middle (LEM), i.e. “either P or not P”. Claimed you didn’t know P unless you could check every case.


• A proof Brouwer wouldn’t believe:• Theorem: There are irrational numbers x and ysuch that is rational.

• Proof: We know that is irrational. If is rational, we have the result. If it is irrational,

then is rational, since


• What’s wrong with this proof from an Intuitionistic viewpoint?

• It assumes is either rational or irrational, which uses the LEM.

• The proof gives no way of determining which of the two alternatives is true.


• Brouwer saw LEM as implying that every proposition could be proved or disproved, and he rejected this implication.

• Brouwer felt numbers and indeed all things that mathematics is concerned with are mental constructions and only exist in our minds.

• Heyting developed an “intuitionistic” logic.


• The major problem with Intuitionism was thatmuch of mathematics was then, and even more is now, based on logical deductive methods that were rejected by Brouwer and his followers. Thus, for example, many existence proofs were not acceptable. Thus, much of modern mathematics can’t be proved constructively.

• It is now mostly an historical artifact and a study for a few mathematical logicians and philosophers.

Hilbert’s Formalism• Hilbert was peeved with the

set‐theoretic paradoxes.• “The present state of affairs

where we run up against the paradoxes is intolerable. Just think, the definitions and deductive methods which everyone learns, teaches and uses in mathematics, the paragon of truth and certitude, lead to absurdities! If mathematical thinking is defective, where are we to find truth and certitude?”

Hilbert’s Plan:

• Provide a proof, from within mathematics, that classical mathematics was consistent (notice there is no mention of truth here), and do so with proofs acceptable to everyone (including Brouwer).

• He was willing to give up “truth” in favor of consistency – getting rid of those darn paradoxes.

Hilbert’s Plan:

• Eventually, this plan was brought down, along with the more general “Formalism” that grew out of it, and along with Russell and Whitehead, by the work of Kurt Gödel.

• But before we talk about that, we need to talk about Formalism as a broader movement.

Formalism

• Mathematics is a game of logical deduction, or “the science of rigorous proof.”

• It begins with axioms and produces theorems. • It rejects the notion of truth and rejects the Euclid Myth. Euclidean and Non‐Euclidean geometries are neither true nor false, but just the result of correct logical deduction from different axiom sets.

• Mathematical statements aren’t about anything and don’t mean anything.

Formalism

• “From the formalist point of view, we haven’t really started doing mathematics until we have stated some hypotheses and begun a proof. Once we have reached our conclusions, the mathematics is over.”

• From the viewpoint of science (especially the logical positivists), mathematics is a language.

• The important thing is consistency – lack of any contradictions.

Formalism

• Nicolas Bourbaki, a group of mathematicians writing under a pen name, became the strongest proponents of mathematical formalism in the 1950’s and 1960’s, creating a series of graduate texts in mathematics using axiomatic developments, with very little or no diagrams or applications.

• I’m pretty much infected with it, as are you.

Troubles with Formalism

• It doesn’t square with what working mathematicians actually do.

• I doesn’t square with what most of us think about mathematical and arithmetical statements.

• And then there’s Gödel.

Incompleteness

• In 1930 – 1931, Kurt Gödel proved that any consistent formal system with enough power to develop elementary arithmetic would have statements that were true, but unproveable.

Incompleteness

• Thus, the system would be incomplete – unable to decide the truth of some statements.

• This is Gödel’s First Incompleteness Theorem.• Gödel’s Second Incompleteness Theorem showed that you cannot prove the consistency of arithmetic from within arithmetic.

• In other words, Hilbert’s plan was doomed to failure.

• It was a depressing time for many mathematicians, apparently.

Incompleteness

• The particular unproveable statement that Gödel created was a sneaky, underhanded, self‐referential monster that was created just for the purposes of the proof.

• But: we have actually come upon some more or less “regular” mathematical statements that are undecidable in that neither they nor their negations can be proved from our set‐theory axioms.

Incompleteness

• The first four axioms of Euclidean Geometry (properly updated and formalized) are incomplete in something like this sense.

• A simple way to think of this is that the first four axioms are true in Euclidean Geometry, but are also true in Hyperbolic Geometry, so that the axioms and any theorems that follow from them aren’t strong enough to decide how parallel lines should behave.

Incompleteness

• One way to show a system is incomplete to create a model in which some statement is true, and another model in which the statement is false.

• This is easy in geometry, but it is much harder when your axiom set describes most of mathematics – like the axioms of set theory.

Incompleteness

• Nevertheless, in our usual set theory, there are many such statements, assuming set theory is itself consistent: CH, GCH, the Axiom of Choice, the Well Ordering Principle, Zorn’s Lemma, the Hausdorff Maximal Principle, Martin’s Axiom, Suslin’s Conjecture, ◊, V=L, etc. etc.

Incompleteness

• Suslin Conjecture: every countable‐chain‐condition dense complete linear order without endpoints is isomorphic to the real line.

• Martin’s Axiom: MA(k) is the statement that for any partial order P satisfying the countable chain condition (hereafter ccc) and any family D of dense sets in P, with | D | at most k, there is a filter F on P such that F ∩ d is non‐empty for every d in D. Martin’s Axiom says that MA(k) holds for every k less than the continuum.

Incompleteness

• In modern set theory there are two methods used to show a statement is unproveable.

• Gödel developed a model of set theory by carefully constructing only those sets which absolutely need to exist to satisfy the axioms. This is called the Constructible Universe, and is denoted by L. When we say V=L, we are invoking the model. It’s the minimal model for set theory, in some sense, and in it the Continuum Hypothesis is true, because it has only those infinite sets that absolutely need to be there.

Incompleteness

• The second method was developed by Paul Cohen at Stanford in the 1960’s. It is called “forcing” and is a method for adding a lot of sets to a model in a way that doesn’t specify too much about them – they are called “generic” sets. He proved that there was a model of set theory with an infinite set of size between that of the integers and that of their power set.

My Two Mathematical Heroes

Together, they proved both CH and AC independent of ZFC!

Incompleteness

• We are doomed to have to make decisions on these things “outside” of the axiom systems in which they appear, since the axioms systems can’t prove or disprove them.

• The axiom systems just ain’t strong enough to decide everything.

• So formalism fails as a foundation for mathematics.

Formalism

• That doesn’t mean it isn’t alive and well, of course, just that it didn’t do what it was originally intended to do.

• Also, most working mathematicians couldn’t care less about this issue, really.

• Remember, they tend to be Platonists on weekdays, and formalists on Sundays.

• But mostly they want to be left alone to do some mathematics.

The Philosophical Plight of the Working Mathematician

• Dieudonné: “On foundations we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes we rush to hide behind formalism and say, ‘Mathematics is just a combination of meaningless symbols,’ and then we bring out Chapters 1 and 2 on set theory. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. This sensation is probably an illusion, but is very convenient. That is Bourbaki’s attitude toward foundations.”


• Cohen: “To the average mathematician who merely wants to know his work is accurately based, the most appealing choice is to avoid difficulties by means of Hilbert’s program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency. . . .The Realist [i.e., Platonist] position is probably the one which most mathematicians would prefer to take.”


• Cohen, continued: “It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism, while his normal position will be somewhere between the two, trying to enjoy the best of two worlds.”


Davis and Hersh summarize this by saying that “…the typical working mathematician is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.”

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