Mathematics as Philosophy

David Pierce 2003, October 27

    http:
//www.math.metu.edu.tr/~dpierce/talks/

AΓEΩMETPHTOΣ MH EIΣITΩ (Geometri yapamayan girmesin):

Motto of the American Mathematical Society, presumably based on Platonic

tradition

Mathematics is justified by its service to philosophy!

FACIO LIBEROS EX LIBERIS LIBRIS

LIBRAQUE

(Teraziyle ve kitaplarla çocuklardan özgür insanlar yaparım):

Motto of St John’s College, Annapolis, Maryland, and Santa Fe, New Mexico, USA.

“The following teachers will return to St John’s College next year:

“Homer, Æschylus, Herodotus, Plato, Sophocles, Aristophanes, Thucydides, Aristotle, Euripides, Lucretius, the Bible,

Plutarch, Virgil, Tacitus, Epictetus, Plotinus, Augustine, Anselm, Aquinas,

Dante, Chaucer, Shakespeare, Machiavelli, Montaigne, Bacon, Descartes, Cervantes, Pascal, Milton, Hobbes, Locke, Rousseau,

Swift, Leibniz, Berkeley, Hume, Kant, Wordsworth, Austen, Smith, Twain, Tolstoy, Goethe, Hegel, Tocqueville,

Kierkegaard, Dostoevski, Marx, Nietzsche, Freud… ”

Science

Mathematics

Model-theory

Logic Philosophy

Mathematical truths are:

like philosophical truths, personal: not needing verification by experiment or by the

agreement of a multitude; but,

like scientific truths, universal: one expects them to be agreed on by all who take the

trouble to understand them—and the agreement generally happens.

Unwilling math-student’s complaint: “There’s only one right answer!”

In model-theory, a model is a structure, considered with respect to some theory or theories of which it is a model.

(Likewise a person is a son or daughter with respect to his or her parents.)

Model-theory is:

mathematics done with an awareness of the language with which one does it;

the study of the construction and classification of structures within specified classes of structures (Wilfred Hodges, Model Theory);

algebraic geometry without fields (Hodges, A Shorter Model Theory);

the geography of tame mathematics (Lou van den Dries);

the study of structures quâ models of theories.

Bourbaki on the role of logic: “In other words, logic, so far as we mathematicians are concerned, is no more and no less than the grammar of the language which we use, a language which had to exist before the grammar could be constructed…

“The primary task of the logician is thus the analysis of the body of existing mathematical texts, particularly of those which by common agreement are regarded as the most correct ones, or, as one formerly used to say, the most ‘rigorous.’” (“Foundations of Mathematics for the Working Mathematician”, Journal of Symbolic Logic 14, 1949)

Is mathematics one?

Some possible divisions:

pure/applied

geometry/algebra/arithmetic/analysis/…

Euclid gives all mathematical facts in geometric terms, even those facts that we

should call arithmetic or algebraic

Descartes: Euclid must have used non-geometrical means to discover some of

those facts

Euclid, Elements II.4: “If a straight line be cut at random, the square on the whole is equal to the squares on the segments and

twice the rectangle contained by the segments.”

Descartes, Geometry: By geometric means, the squares and rectangles can be replaced by straight lines, once we have chosen a

straight line to represent unity.

Thus, all arithmetic operations can be referred to geometry—though there may be

no practical need to do so.

Old-style geometry is not abandoned: see Lobachevski.

A possible analogy:

Algebra, geometry &c. are to the mathematical world as the senses are to the

physical world.

We have a common sense whereby the things that we see, hear and touch are

known to be part of one world.

The visible, audible and tangible worlds are the same in principle, but it would be foolish to treat the world as simply one of these.

We might even hope to develop additional senses, the better to understand the world:

THERE is NO NATURAL RELIGION (William Blake, c. 1788):

The Argument: Man has no notion of moral fitness but from Education. Naturally he is only a natural organ subject to Sense.

I Man cannot naturally Percieve, but through

his natural or bodily organs II Man by his reasoning power. can only

compare & judge of what he has already perciev’d.

III From a perception of only 3 senses or 3

elements none could deduce a fourth or fifth IV None could have other than natural or

organic thoughts if he had none but organic perceptions

V Mans desires are limited by his perceptions.

none can desire what he has not perciev’d VI The desires & perceptions of man untaught

by any thing but organs of sense, must be limited to objects of sense.

“The essential business of language is to assert or deny facts.” (Bertrand Russell, introduction to Wittgenstein’s Tractatus, 1922)

“Bodily actions expressing certain emotions, in so far as they come under our control and are conceived by us, in our awareness of controlling them, as our way of expressing these emotions, are language… The grammatical and logical articulations of intellectualized language are no more fundamental to language as such than the articulations of bone and limb are fundamental to living tissue” (R.G. Collingwood, The Principles of Art, 1937).

What is an assertion of a fact?

Say it is an utterance, in a certain manner, of a statement (or proposition).

What is a statement?

Some authors say it is a sentence that is either true or false.

I think a statement is a sentence distinguished from other sentences by its form: it is not a question, but its verb is declarative, not subjunctive or imperative, …

Then a statement becomes true or false when placed in an appropriate context.

A true statement (in context) need not be a correct answer to the question it is intended to answer.

Bourbaki: What are the structures at the heart of mathematics? They can be:

algebraic, like:

the fields (ℝ, +, ×) and (ℂ, +, ×), the vector-spaces ℝ^{n} and ℂ^{n}, the group of rigid motions of the Euclidean plane …;

ordered:— totally, like (ℤ, ⩽) and (ℝ, ⩽), or partially, like (𝒫(Ω), ⊆), …;

topological: the real number x is in the interior of the set A if: for some positive ε, if |y − x| < ε, then y is in A; symbolically,

∃ε > 0 ∀y (|y − x| < ε ⇒ y ∈ A)

Bourbaki aims to characterize structures by means of axioms.

Model-theory treats directly of the first two kinds of structures and examines how axioms fail to characterize any one structure.

Let ω be the set of natural numbers: the smallest of sets Ω of sets such that ∅ ∈ Ω and, for all sets A, if A ∈ Ω, then A ∪ {A} ∈ Ω. Let j and n range over ω.

We write 0 for ∅, and 1 for {0}, and so forth; also n + 1 for n ∪ {n}. If j ⊂ n, we write j < n. Then n = {j : j < n}.

Let I be a finite subset of ω. The Cartesian power M^{I} is the set of functions from I to M; a typical element a of M^{I} can be written (a_{j} : j ∈ I).

Since 0 is empty, M^{0} consists of the empty function, 0; so M^{0} = {0} = 1.

M^{n} consists of (a_{0}, …, a_{n−1}), where a_{j} ∈ M.

An I-ary operation on M is a function from M^{I} to M.

Atn I-ary relation is a subset of M^{I}.

A structure on M is a set of operations and relations on M, each of them n-ary for some n in ω.

In the background are three “primordial” structures:

• the structure of the natural numbers:

(ω,′, 0),

where ′ is the unary operation x ↦ x + 1;

• the Boolean algebra of subsets of a “universal” set U:

(𝒫(U), ∩,^{c}, ∪, ∅, U, ⊆);

• propositional logic:

(𝔹, ∧, ¬, ∨, 0, 1, ⇒),

where 𝔹 = {0, 1} = 𝒫({∅}) and can be understood as {false , true }.

A structure on M may be denoted 𝔐. This has a signature: a set L of symbols for the specified operations and relations.

The symbols in L are primary; say they are f and R, symbolizing n(f)-ary operations f^{𝔐} and n(R)-ary relations R^{𝔐} on M.

Any element a of M can be understood as a nullary operation-symbol: a constant-symbol. For any subset A of M, for the signature L ∪ A there is a first-order language, L_{ωω}(A).

For an I-ary formula ϕ of L_{ωω}(A), there is an I-ary relation ϕ^{𝔐} on M.

Such relations are the A-definable sets of 𝔐; they compose Boolean sub-algebras 𝒟^{I}(A) of the 𝒫(M^{I}).

SYMBOL INTERPRETATION
SYNTAX SEMANTICS
signature L 𝔐, an L-structure

OPERATIONS ON M:

variable x_{j} a ↦ a_{j} : M^{I} → M
constant-symb. c  c^{𝔐}, an element of M
function-symb. f  f^{𝔐} : M^{n(f)} → M
term t t^{𝔐}, a composition
LOGICAL: FUNCTIONS ON 𝒫(M^{I})
connectives:  operations:
∧ ∩
¬ A ↦ A^{c}
∨ ∪
→ (A, B) ↦ A^{c} ∪ B
quantifiers:  projections:
∃x_{j} π_{j}^{I}: a ↦ (a_{i} : i ∈ I \ {j})
∀x_{j} A ↦ (π_{j}^{I}(A^{c}))^{c}

RELATIONS ON M:

equals-sign =  equality
relation-symb. R  R^{𝔐}, a subset of M^{n(R)}
I-ary formula ϕ ϕ^{𝔐}, a subset of M^{I}

nullary formula

or sentence σ 1 (true) or 0 (false)

One may care only about the definable sets:

Let also J be a finite subset of ω. If

σ : I → J,

then σ^{∗} : M^{J} → M^{I}, whence

σ^{∗} : 𝒫(M^{J}) → 𝒫(M^{I}),
σ_{∗} : 𝒫(M^{I}) → 𝒫(M^{J}),

where σ^{∗}(c) = (c_{σ(i)} : i ∈ J) if c ∈ M^{J}, and

σ^{∗}A = {σ^{∗}(c) : c ∈ A},
σ_{∗}B = {c : σ^{∗}(c) ∈ B}.

We can re-define a structure on M to be 𝒟, which comprises, for each I, a Boolean sub-algebra 𝒟^{I} of 𝒫(M^{I}) such that

{(c, c) ∈ M^{2}} ∈ 𝒟^{2};
A ∈ 𝒟^{J} ⇒ σ^{∗}A ∈ 𝒟^{I};
B ∈ 𝒟^{I} ⇒ σ_{∗}B ∈ 𝒟^{J}.

From 𝒟 we can recover a structure on M in the original sense whose 0-definable sets are just the sets in the 𝒟^{I}.