Mathematical Proof
History and Philosophy of STEM
Lecture 24
Chess Problems
Euclid on the Infinity of Primes
Every number is either prime, or divisible by a prime. Assume that
there are a finite number of primes, and let P be the last prime
number
Q = (2 ⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ ⋅ ⋅ P) + 1
Is Q prime, or not? If it is, then P can’t be the last prime, because Q is
bigger than P. If it isn’t, then there has to be some prime bigger than
P, because Q isn’t divisible by any of the primes in that list. On either
supposition, there is a prime larger than P, which is a contradiction.
QED
Pythagoras on Irrationals
√
If 2 is rational, then it can be written as:
√
2 = ab ,
where a and b are chosen such that you can’t reduce that fraction any
more. You can thus write this as:
2
2 = ab2
a2 = 2b2
Pythagoras on Irrationals
a2 = 2b2
Obviously a2 is even, since it’s expressible as 2x. Whenever you
square an odd number, you get another odd number. So a must be
even. That means
a = 2c
for some value of c. Plug that in, and you get:
Pythagoras on Irrationals
(2c)2 = 2b2
4c2 = 2b2
2c2 = b2
Once again, since b2 is even, that means that b is even. So we have it
that both a and b are even. But wait! A few slides ago, we said that
the fraction ab was as reduced as possible. If both a and b are even, that
means we can cancel a 2 from √
the fraction. So that’s a contradiction!
You must not be able to write 2 that way after all, which means it’s
irrational. QED
Mathematical Knowledge
What kind of knowledge is this?
How are proofs generating knowledge?
Mathematical Objects
• Platonism: mathematical objects are abstract objects, “out
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there” in Platonic heaven; we discover the truths of mathematics
Formalism: mathematical objects are defined by games we
play using mathematical symbols; we invent the truths of
mathematics
Intuitionism: mathematical objects are constructed by doing
proofs from logical basics (invent)
Structuralism: mathematical objects aren’t really there, but the
structures that they form parts of are (discover-ish)
Empiricism: mathematical truths are derived from ordinary
experience, just like science (discover)
Fictionalism: mathematical truths are just useful fictions we
tell ourselves (invent)
Characteristics of Mathematical
Knowledge
necessity – mathematical knowledge, properly derived, could not
have been otherwise
beauty – really hard to define, but still easy even for novices to
recognize
seriousness – separates real mathematics from chess problems;
something like the significance of interconnectedness of
mathematical concepts
Seriousness
• Euclid’s prime proof: shows there’s an infinity of number theory
work to do, we’ll never “run out” of primes
√ √ √
3, 5, 3 17,
etc.; shows there’s an infinity of irrationals that don’t work like
the rationals (actually, lots more irrationals than rationals)
• Has nothing to do with practical applications
• Pythagoras’s irrational proof: easy to extend to
Approximating Pi
Susan Gomez, manager of the International Space Station
Guidance Navigation and Control (GNC) subsystem for NASA,
said that calculations involving pi use 15 digits for GNC
code and 16 for the Space Integrated Global Positioning System/Inertial Navigation System (SIGI). SIGI is the program that
controls and stabilizes spacecraft during missions. (Scientific
American)
Answers?
Nope. This is a really hard problem.
Questions?
• How similar is this to scientific knowledge?
• Pros:
• Remember explanation as unification
• Empiricist approach to mathematics makes them very similar
• Cons:
• A Platonist picture makes them seem very different