Today:
Reading questions.
For next time:
Reading questions below.
Prepare for a quiz.
Start Homework 9.
From Hofstadter, Godel, Escher, Bach
Symbols: M, I, U
Axiom: MI
Rules:
1) xI -> xIU
2) Mx -> Mxx
3) xIIIy -> xUy
4) xUUy -> xy
Is MU in the theorem?
Mathematical Realism:
R1: mathematical entities exist independent of the human mind
R2: mathematical truths are discovered
R3: the same truths would be discovered by any mathematicians
Canonical proponent: Plato
Intuitionism:
I1: mathematical objects are a construction of the mind
to say that an object exists, there has to be a constructive process for creating it; not sufficient to refute
its non-existence
I2: to say that a statement is true is exactly to say that it can be proved
reject the law of the excluded middle, since it is not always true that either A or not-A can be proved
generally don't accept proof by contradiction
also not wild about the axiom of choice and infinity.
I3: mathematical objects exist in the mind
language can be used to help another person reconstruct an object, but the language has no official role
Canonical proponent: Luitzen Egbertus Jan Brouwer, known to his friends as Bertus
Formalism:
F1: mathematics is the study of formal axiom systems (symbols, axioms, and rules for manipulating strings of symbols)
F2: contrary to I3, the product of mathematics is the language, not the brain state
F3: in the strong form, mathematics is _just_ string manupulation
in the deductivist form, you can attach meaning to mathematical statement if you accept the axioms and a mapping of the symbols onto something in the world
F4: "true and false" apply only at the semantic level
"provable and not provable" apply only at the symbol level
Canonical proponent: David Hilbert
Positivism:
"a severe theory of meaning that makes liberal use of the word 'meaningless'"
--- Goldstein, Incompleteness
P1: Mathematical statements are logical consequences of their axioms, and are meaningless.
A mathematical equation simply says that the meaning of both sides is the same.
P2: A descriptive statement (as opposed to a mathematical one) without a verification process is meaningless.
P3: The meaning of a descriptive statement is that the verification process would be successful.
Canonical proponent: Ludwig Wittgenstein, who may or may not have menaced Karl Popper with a fire poker.
1900 version: (#2 on the list presented at the Paris conference of the
International Congress of Mathematicians)
Find a provably consistent FAS that derives all true
statements of arithmetic (of non-negative integers)
1920 version: a provably consistent formal system that could
derive all true statements in all mathematics
Clarification: completeness is only meaningful if you are trying to attach semantics to a FAS.
The association of a FAS with semantics is called an interpretation.
If a string A can be derived from the axioms, it is derivable, provable, or "in the theorem"
A FAS "maps onto a semantic system (Allen's language)" if
1) there exists an interpretation such that every derivable string A maps to a statement Interp(A) that is true in the semantic system
Example: consider a FAS with only one axiom, the ASCII string '49 43 49 61 50', and no generative rules.
If we use the interpretation
49 -> 1 43 -> + 61 -> = 50 -> 2
Then the axiom maps to the statement 1+1=2. Assuming that we also apply the usual interpretation of numerals and symbols, this is a true statement in the semantic system we call arithmetic.
So it is easy to come up with any number of FASs that map onto a given semantic system.
It's harder to find one that's complete.
A FAS+Interpretation is complete if
1) the Interpretation maps the FAS onto a semantic system
2) every true statement in the semantic system maps to a string that is derivable
This is what Hilbert was asking for, but in addition, the FAS+Interp had to be provably consistent:
A FAS+Interp is consistent if there is no pair of strings, A and B, in the theorem that map to statements Interp(A) and Interp(B) that are contradictory (in the semantic system).
.
Test your understanding
Question: if an FAS derives a statement A, and Interp(A) is false, what does that tell us about the consistency and/or completeness of the FAS?
Question: how could you possibly prove consistency?
In general, I don't know. But in a system that contains basic logic, it is provable that p -> (~p -> q)
In words, this means that if we can derive p and ~p for any p, then we can derive any string q.
The contrapositive of this statement is that if there is an string q that we cannot derive, then there is no pair (p, ~p) that can be derived.
So, in a system with basic logic, you can prove consistency just by finding one string that can't be derived!
Well, how do you do that?
To understand incompleteness, it is useful to keep separate (at least for now) three levels:
1) syntax: strings of symbols.
(∀y)(∃x)(x=Sy)
2) semantics
Under the interpretation called Peano Arithmetic, the
string '(∀y)(∃x)(x=Sy)' denotes the statement
"For all y there exists an x such that x is the successor of y"
3) meta-math
Meta-mathematical sentences include:
a) "(∀y)(∃x)(x=Sy)" is a string in PA
b) "(∀y)(∃x)(x=Sy)" is a derivable string in PA
c) In PA, the string '(∀y)(∃x)(x=Sy)' denotes the statement
"For all y there exists an x such that x is the successor of y"
d) PA is consistent
e) PA is complete
f) In MIU, there are no derivable strings with n Is and n divisible by 3.
Therefore MU is not a derivable string.
What you should read:
Casti, Five Golden Rules, Chapter 4 (handout attached)
pages 152 (starting with "Form and Content") to 174 (ending at "Tough times")
1) What was the ultimate philosophical goal of the challenge Hilbert posed?
2) What is a formal system? What is "the theorem" of a formal system?
3) What is a metamathematical property?
4) What does the "dictionary" in Figure 4.4 do?
5) What does it mean to say that a formal system is complete?
6) The first complete sentence on the top of page 158, beginning "And you certainly can't," is nonsense. Why?
7) What is the Big Idea #1 that underlies Godel's proof?
8) What is Big Idea #2?
The statement of Godel's Theorem on page 163 is called the First Incompleteness theorem.
The Second Incompleteness Theorem takes on the second part of Hilbert's challenge, provable consistency.
9) What is the Turing Test?
10) What is the basic claim Penrose makes about the limitations of machines?
11) Casti outlines three responses to Penrose and Lucas. What are they?
12) What is the smallest dull number?
13) What is the definition of omega?
14) How do we know that we can't compute omega?