Mathematical Aims Beyond Justification
Fiona T. Doherty
The Method is not the Message: The Frege-Hilbert Controversy
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Frege: Existence → Consistency
For Frege axioms are truths, i.e. the propositions expressed by determinate true sentences.
• Consistency: Propositions are consistent if they can be true together.
• Syntactic consistency: Sentences are consistent if you cannot derive a contradiction from
them, i.e. if ⊥ is not a member of their deductive closure.
• Property consistency: A property is consistent if it is satisfiable. “...just in case some
series of concepts or sets could have that property” (Blanchette, 1996:320-322).
“What means have we of demonstrating that certain properties... do not contradict
one another? The only means I know is this: to point to an object that has all
those properties, to give a case where all those requirements are satisfied. It does
not seem possible to demonstrate the lack of contradiction in any other way” (Frege
to Hilbert, 1980:43).
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Hilbert: Consistency → Existence
“If they arbitrarily given axioms do not contradict one another with all their consequences, then they are true and the things defined by the axioms exist. This is the
criterion of truth and existence” (Hilbert to Frege, 1980:39-40).
Frege protests:
“Our views are perhaps most sharply opposed with regard to your criterion of existence and truth...Suppose we knew that the proposition
(1) A is an intelligent being
(2) A is omnipresent
(3) A is omnipotent
together with all their consequences did not contradict one another; could we infer
from this that there was an omnipotent, omnipresent, intelligent being? This is not
evident to me.” (Frege to Hilbert, 1980:47)
3
Hilbert’s Methodology
Guide to Hilbert style consistency proof:
1. Take a set of sentences and identify the non-logical terms in its members.
2. Reinterpret all of its non-logical terms using the background theory.
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3. Prove that so interpreted, the sentences are theorems of the background theory, i.e. they
are true.
Example of step (2):
“The axioms of this group [connection] establish a connection between the concepts
indicated above; namely, points, straight lines, and planes. These axioms are as
follows:
[I, 1.] Two distinct points A and B always completely determine a straight line a.
We write AB = a or BA = a.
[I, 2.] For every two points there exists at most one line on which lie between those
points. ...(Hilbert, 1899:3-4)
[I, 2.] For every two points there exists at most one line on which lie between those points.
• P is a point; assigned the set of pairs hx, yi of Reals
• L is a line; assigned the set of ratios [u : v : w] of Reals
• P lies on L; assigned the set of pairs {hx, yi, [u : v : w]}such that ux + vy + w = 0.
[I, 2.R ] For any pair of pairs of real numbers hha, bi, hc, dii there is at most one ratio of real
numbers [e : f : g], such that both ae + bf + g = 0 and ce + df + g = 0.
3.1
Frege’s diagnosis
“The characteristic marks you give in your axioms are apparently all higher than
first-level; i.e., they do not answer to the question “What properties must an object
have in order to be a point (a line, plane, etc.)?”, but they contain, e.g., second-level
relations, e.g., between the concept point and the concept line. It seems to me that
you really want to define second-level concepts but do not clearly distinguish them
from first level ones.” (Frege to Hilbert, 1980:46)
Hilbert’s axioms define a complex second-level predicate in which his non-logical
primitives are not themselves furnished with meaning but instead function as variables marking the argument places in a single six-place higher-order relational
expression. This expression refers to the concept of being a Euclidean Space.
3.2
Hilbert’s reaction
“It is surely obvious that every theory is only a scaffolding or schema of concepts
together with their necessary relations to one another, and that the basic elements
can be thought of in any way one likes. If in speaking of my points I think of some
system of things, e.g. the system: love, law, chimney-sweep... and then assume all
my axioms as relations between these things, then my propositions, e.g. Pythagoras’
theorem, are also valid for these things. In other words: any theory can always be
applied to infinitely many systems of basic elements.” (Hilbert to Frege, 1980:40-41)
[I, 2.*] In any system S, for two distinct objects a, b ∈ S that are points-in-S then there is
exactly one third object c that is a line-in-S and which lies-between-in-S the points a and b.
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