Where are we at?
- Galileo provided an example of both the dawning
power of formal reason (geometry as the foundation of
two new sciences), but also the conundrums that
remained concerning some kinds of reasoning (e.g.,
- Cantor formalized what Galileo said could not be
- But, Cantor's system has a paradox!
- Other prominent concerns include the development
of non-Euclidean geometry. If we learn about geometry
from our intuitions of space, and these led us to see
the necessary truths of Euclidean geometry, then why
do we also see the necessary truths of non-Euclidean
- Consensus developed that we need to step back,
return to the basics
- Next: foundationalisms, the algorithm, and
paradoxes and limits.
Review of Cantor's Antinomy
- Two things contradict: each set is smaller than
its power set, but there is a set of all sets (which
is thus the largest set).
- Some smart class made two suggestions:
1. Could we keep unrestricted set
formation but rule out just the
universal set? There are systems like
this, but this move alone will not
work: as we'll see with our discussion
of Frege, paradoxes can still be
2. Can we rule out self-reference?
This is interesting, and we need to
consider if this is possible. Frege
tried it, however, and it was later
found to not rule out other paradoxes;
approaches like this that do work include
Russell's theory of types.
- What is done in set theory today? Briefly pure
set theory requires that someone prove a set "exists"
by "constructing" it (by describing a procedure that
gets you to that object out of arrangements of other
objects previously proven to exist or accepted to
exist). To foreshadow: such an approach rules out the
ability to just derive most mathematical principles --
rather, with such a principle, one must explicitly
posit them as "axioms" (an axiom is a principle or law
that one starts with, and does not derive.)
- Issues like the invention of non-Euclidean
geometry, the controversies over infinity, and concern about
calculus and real numbers, led to a deep concern about the
foundations of reason.
- Since our intuitions about what one can and cannot reason
about conflict (e.g., Kronecker versus Cantor), can we get rid
- Three different approaches to exploring foundations
1. Logicism: sound mathematical reasoning, and
the admissible entities of mathematics, will be
reducible to logic.
2. Formalism: sound mathematical reasoning is either
reducible to logic or is a description of the
manipulation of real world objects (symbols).
3. Intuitionism: sound mathematical reasoning, and the
admissible entities of mathematics, do not need
justification, but are given in a special intuition
(and through constructions based upon those
- (To this list, we are going to add before the end
of the semester another kind of reasoning about
reasoning, a semi-empirical approach.)
Frege and Logicism: Frege background
- Born 1848 in Wismar, Germany
- Both his parents were educators
- Studied at Universities in Jena and Gottingen
- Received doctorate in 1873
- Begriffsschrift (Concept Script) published 1879
- Grundgesetze der Arithmetik (Basic Laws of Arithmetic), Volume 1 published 1893; volume 2 published 1903.
- Russell communicates the paradox in 1902.
- Later volumes of the Basic Laws attempt to avoid the paradox, but fail.
- Frege died, believing that his program had failed, in 1925
- Frege's goal is to expunge reference to
intuitions from logic, by reducing all of mathematics
to logic (which means at least, very general and
- In his Foundations of Arithmetic, Frege
attacks psychologism (the view that mathematics can be
explained via psychology), and asks "what is a
number?" His answer is a good first step towards his
- Frege argues that we can understand the
number of things falling under a concept F by
way of the more general concept of being
- Thus, we use the idea that (1) the
extension of the concept F is equinumerous
with the extension of the concept G, to
make sense of the idea that (2) the number
of Fs is the same as the number of Gs.
- Frege says, "The Number that belongs to
the concept F is the extension of the concept
'equinumerous to the concept F'"
- Here is then how he defines zero: The
number 0 is the Number which belongs to the
concept not identical with itself
- The number 1 is the Number that belongs
to the concept identical with 0.
- The number 2 is the Number that belongs
to the concept identical with 0 or 1.
- The number n+1 is the Number which belongs
to the concept member of the natural number
series ending with n.
- In his Begriffsschrift, Frege developed
single-handedly the system which underlies modern
logic. His many achievements include the invention of
- In the Basic Laws of Arithmetic, Frege
sought to apply and extend the system of the
Begriffsschrift, and show that from this he
could derive arithmetic.
- In this system, one
basic principle is that a concept has a referent (or,
as Frege would say, it has an extension) if it is
constructed out of other concepts using basic logical
principles, and we know those other concepts have a
- Frege did not note, however, that his system has
at least one necessary exception to this principle,
"Law V" (the fifth law posited in Basic Laws of
Roughly, in English, this law says:
Whatever satisfies a concept F also satisfies
the concept G, and vice versa, if and only if
the concepts F and G have the same extension.
- Law V is an instance of what logicians now call
"second order logic"
It is helpful to see the axioms of his system. Roughly, in
contemporary formulation, and followed by both an english
"translation" and also a very rough example, are the axioms. Let "A"
stand for the universal quantifier, all. Let P, Q be sentences (these
are each either true or false, but not both and not neither). Let F
and G be predicates (these label properties). Let a be some
Axiom 1 a and b:
a. (P → (Q → P))
b. (P → P)
"Translation": If sentence P is true, then if sentence Q is true,
P is true.
Example: If the streets are wet, then if it rains
then the streets are wet. It's not obvious at first why such a
principle is useful, but it allows you to do many important
things, like prove a conditional.
(∀xFx → Fa)
"Translation": If everything has property F, then some particular
a has property F.
Example (assume we're talking only about
numbers): If every number is divisible by 1, then 15 is divisible
(∀F(Fa) → Ga)
"Translation": If a has every property, then a has
particular property G.
∀x∀y(x=y → ∀F(Fx → Fy))
"Translation": If x and y are the same, then any property x has
is also had by y.
Example: If number x is the same as number
y, then if x is divisible by 4, y is divisible by 4.
¬(P ↔ ¬Q) → (P ↔ Q)
"Translation": if it is not the case that sentence P is true just in case Q is false, then P is true just in case Q is true.
Basically, this says that if it is not the case that P and Q have different truth values, then they have the same truth value.
((x'Fx = x'Gx) ↔ ∀x(Fx ↔ Gx))
"Translation": the collection of things that F is true of is the
same collection of things that G is true of, just in case for any
thing x, x has property F if and only if x has property G. For
example, the collections of humans is the same as the collection
of rational animals, just in case if all and only humans are
a = ie'(a = e)
"Translation": object a is the thing with property e such
that a is the sole extension of e. This basically means that each
object will have a unique description true only of it. Thus,
suppose a stands for Abraham Lincoln. Then: a = the
thing of the extension e where e is the property of being the 16th
President of the United States and a is e.
Frege needs Axiom 5 to develop the elements of mathematics that he
aims to develop. However, he assumes that every concept is defined
over every object. Axiom 5 also allows that the extension of a
concept is an "object" -- it is a thing that we can quantify over and
treat the way we treat other objects, like numbers, of our theory.
Finally, Axiom 5 has as a consequence that once we formulate a concept
it has an extension. These facts allow as concepts with extensions
things like "the concept of being a concept that is not satisfied by
- Frege received from Bertrand Russell a letter of
June 16, 1902, which said in part:
.... I have known your Grundgesetze der
Arithmetik for a year and a half, but only
now have I been able to find the time for the
thorough study I intend to devote to your
writings. I find myself in full accord with
you on all main points, especially in your
rejection of any psychological element in
logic and in the value you attach to a
Begriffsschrift for the foundations of
mathematics and of formal logic, which,
incidentally, can hardly be distinguished. On
many questions of detail, I find discussions,
distinctions and definitions in your writings
for which one looks in vain in other
logicians. On functions in particular
(Section 9 of your Begriffsschrift) I
have been led independently to the same views
even in detail. I have encountered a
difficulty only on one point. You assert
(p. 17) that a function could also constitute
the indefinite element. This is what I used
to believe, but this view now seems to me
dubious because of the following
contradiction: Let w be the predicate
of being a predicate which cannot be
predicated of itself. Can w be
predicated of itself? From either answer
follows its contradictory. We must therefore
conclude that w is not a predicate.
Likewise, there is no class (as a whole) of
those classes which, as wholes, are not
members of themselves. From this I conclude
that under certain circumstances a definable
set does not form a whole. (Quoted in Beaney
- Frege's reply, written likely the day he
received Russell's note (June 22 June 1902):
.... Your discovery of the contradiction has
surprised me beyond words and, I should almost
like to say, left me thunderstruck, because it
has rocked the ground on which I meant to
build arithmetic. It seems accordinly that
the transformation of the generality of an
equality into an equality of value ranges (set
9 of my Grundgesetze) is not always
permissible, that my law V (section 20, p. 36)
is false, and that my explanations in section
31 do not suffice to secure a Bedeutung
for my combinations of signs in all cases. I
must give some further thought to the matter.
It is all the more serious as the collapse of
my law V seems to undermine not only the
foundations of my arithmetic but the only
possible foundations of arithmetic as such.
And yet, I should think, it must be possible
to set up conditions for the transformation of
the generality of an equality into an equality
of value-ranges so as to retain the essentials
of my proofs. Your discovery is at any rate a
very remarkable one, and it may perhaps lead
to a great advance in logic, undesirable as it
may seem at first. (Quoted in Beaney 1997:
- Russell's paradox is very general (it does
not refer, for example, to the biggest
Cardinal, or other such weird things). Frege
was unimpressed by the Cantorian antinomy, but
considered Russell's paradox devastating.
- Frege himself abandoned logicism, but
Russell continued the program.
I've quoted Frege from the Foundations of Arithmetic,
translated by J. L. Austin, and quoted here out of The Frege
Reader, Michael Beaney (ed.), 1997, Oxford: Blackwell publishers.