1. Philosophy of Mathematics
Gottlob Frege & Logicism

2. Know Your Number System!
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!

3. Part 1 Basics of Frege’s approach

4. Grounds of Knowledge Kantian – three bases for knowledge
Observation [of actuality - Wirklichkeit]
Spatial and temporal intuition
how things must be if we are to apprehend or imagine them as being in space or time [
how things appear to us - to be objects at all
'geometrical truths govern the domain of the spatially intuitable, whether actual or imagined’
Logical faculty
To be thinkable
Arithmetic
paragraph 14, Foundations of Arithmetic 1884
Grounds of Knowledge

5. Arithmetic is a branch of logic
No ground or proof is drawn from intuition or empirical observation
Number system (for natural and real numbers) is to be constructed from:
Definitions – which give meanings to terms
Laws of logic
Axioms must express logical truths
Theorems are then derived from those axioms
Arithmetic is a branch of logic

6. Part 2: Why Mill is wrong

7. Mill is wrong about application
“Mill always confuses the applications that can be made of an arithmetical proposition, which are often physical and do presuppose observed facts, with the pure mathematical proposition itself. The plus sign may indeed seem, in many applications, to correspond to a process of aggregation. But that is not its meaning: for in other applications there is no question of heaps or aggregates, or of the relation of a physical body to its parts, for example when the calculation relates to events.” §9, Foundations of Arithmetic
Mill is wrong about application

8. Mill: “we may call ‘Three is two and one’ a definition of three; but the calculations that depend on that proposition follow not from the definition itself but from an arithmetical theorem presupposed in it, namely that there are collections of objects which while they impress the senses thus:
can be separated into two parts thus:
“This proposition being granted, we call all such parcels ‘threes’; and then the statement of the above-mentioned physical fact will serve also as a definition of ‘three’.”
Frege: “What a mercy, then, that not everything in the world is nailed down; for if it were we should not be able to bring off this separation, and would not be 3! What a pity that Mill did not also illustrate the physical facts underlying the numbers 0 and 1!”
§7, Foundations of Arithmetic
Mill on Numbers

9. Induction & abstraction as processes by which we come to acquire concepts
Pedagogical or psychological
Meaning of those concepts
‘abstraction may help us come to grasp certain concepts - but it has no power to create abstract objects’
Frege: It is not possible to reach a true understanding of numbers based on induction or abstraction
In particular, numbers are not abstract mental constructions conceived by removing properties
Numbers are not homogenous units
Different numbers have different properties
Position in a number series is not the same as position in space
Mill’s Confusion

10. Part 3: Why Formalism is wrong

11. What is Formalism? ‘Game Formalism’ ‘If ... Then’-ism
Axioms are akin to rules of a game
Rules of inference or laws of logic describe the legitimate moves that can be made
Valid Theorem – ‘a certain statement can be obtained from certain other statements by means of certain processes of manipulation’
‘If ... Then’-ism
Bertrand Russell – mathematics is the investigation of conditional statements involving variables
Formalism is not concerned with the “Truth” of mathematical statements nor with the existence of mathematical objects
What is Formalism?

12. Why Formalism is Wrong Frege :
Formalism cannot account for applications of mathematics
'It is applicability alone that raises arithmetic from the rank of a game to that of a science. Applicability therefore belongs to it of necessity.‘
That arithmetic has this kind of sense means its axioms cannot be arbitrary
Formalism can offer no coherent explanation of an infinite sequence such as the decimal expression of a real number (there are only ever finite sequences of figures)
Why Formalism is Wrong

13. Application’s Inflections
Application is integral of the definition of numbers But it is not the sense of mathematical propositions Michael Dummett: 'It is when he is criticising empiricism that Frege insists on the gulf between the senses of mathematical propositions and their applications; it is when he criticises formalism that he stresses that applicability is essential to mathematics.'
Application’s Inflections

14. Part 4 What are numbers for Frege?

15. Logical Objects Independent, but abstract objects
Not actual
But objective – same for all human beings
What we can judge by common criteria must be objective
Realism – mathematician discovers what already exists and gives it a name
Numbers are not subjective, mental creations
‘platonism’ with small ‘p’
All numbers exist
They are given to thought, but are not created in thought
Logical Objects

16. Summary of Frege’s Philosophy of Maths
Dummett: "He [Frege] can happily assert all four of the following propositions:
that the laws of arithmetic can, by means of definitions, be derived by purely logical means from the fundamental laws of logic;
that, in giving those definitions, we must be faithful to the received senses of arithmetical expressions;
that our definitions must completely fix the identity of the natural numbers as specific objects;
that the received senses of numerical terms do not impose any one specific identification of the natural numbers."
Summary of Frege’s Philosophy of Maths

17. Foundations of Arithmetic gives good grounds for thinking that numbers are objects (rather than properties or relations or concepts)
But does not establish that this is the case
Frege’s later, multi-volume work Grundgesetze [Basic Laws of Arithmetic] attempted to provide the proof
But hit a snag ... a contradiction was discovered by Russell
In 1906, Frege abandoned his approach to arithmetic and decided he had been led astray by set theory
A Big Problem for Frege

18. Principle of Comprehension
Naïve set theory includes the Principle of Comprehension.
Anything is a set that can be described by a suitable formal language that can be interpreted as making consistent statements about sets and their elements.
for any formula φ(x) containing x as a free variable, there exists the set {x: φ(x)} with members: those objects that satisfy φ(x).
if the formula φ(x) stands for “x is prime”, then {x: φ(x)} will be the set of prime numbers.
If φ(x) stands for “~(x = x)”, then {x: φ(x)} will be the empty set.
This was, broadly speaking, Frege’s approach in his attempts to express arithmetic purely in terms of set theory (actually it’s more complicated than that, but we’re close enough).
Principle of Comprehension

19. 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

20. Part 5 Logicism in Russell & Whitehead’s Principia Mathematica

21. 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

22. Theory of Types 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

23. 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

24. Failure of Logicism? 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
Always at least one sentence that is true but unprovable
No formal system can prove its own consistency
Gödel: Principia Mathematica ‘a considerable step backwards as compared with Frege’ (1944)
Failure of Logicism?

25. ‘[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

26. Part 6 Concluding comments

27. 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:
Gödel - special arithmetical intuition
Empiricism or Naturalism – dispense with the special status of arithmetic & mathematical objects Quine – maths is subordinate to physics
Intuitionism – classical logic & mathematics is a problem
Ways forward

29. Express statements about infinite objects as finite strings of symbols in a precisely-defined formal language.
Express rules of inference as precisely-defined operations on symbol strings.
Derive theorems from the axiom strings by finitely many applications of rules of inference.
Prove that the rules of inference produce no contradictory sentence, by purely finite reasoning about strings of symbols.
Hilbert’s Programme

30. 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.

32. 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

33. 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

34. We may now prove “Humans are fish” as follows:
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.

35. A Version of the ZF Axioms
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

36. 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

37. Predicate logic allows us to “break apart” propositions a bit
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